The direct answer to what you asked is no, there's no way to do that. The cleaning cycle will clear some, even most, clogs, but it isn't a panacea. In fact, some printers will automatically stop running repeated cleaning requests after a preset number of times, and give you a message that further cleaning cycles aren't likely to help. Running cleaning cycles on a single color isn't included as an option because if you get to the situation you describe, it probably won't help, anyway. However, you can likely fix your printing problem another way.
When built-in cleaning cycles don't work
The cleaning cycle relies on the solvent in the ink to dissolve the dried ink. However, that is more ink. If it doesn't quickly dissolve the clog, it can build up and even make things worse. The cleaning cycle exposes the clog to solvent for a few seconds. Stubborn clogs can take orders of magnitude more time. The fix is to actually clean the printhead. I'll describe several procedures you can use to do that.
You basically need to expose the clog to an ink solvent long enough for it to dissolve. Part of the fix is using a procedure to accomplish that, and part is the solvent you use. I'll start with the solvent.
There are many things that will dissolve ink, but doing it in and on the printhead has special considerations. The nozzle holes are microscopic, and any impurities in the cleaner can clog them. Some chemicals can react with chemicals used in some inks and cause strange things, like making residual ink gel. Tap water can be corrosive to the printhead, and contains minerals that can eventually clog the nozzles when residual is evaporated in the printing process.
There are printhead cleaning solutions you can buy and various concoctions you can make yourself (but don't trust every YouTube video that shows how to mix somebody's home-brewed cleaner from household ingredients).
I've successfully used distilled water (sold in gallon jugs in the grocery store for use in ironing, etc.), isopropyl alcohol (the common 70% is fine), and various commercial cleaners. I found that the commercial cleaners tend to do a faster and better job. If you have the time to purchase it, that's the way to go, and then keep it on-hand for next time. In a pinch, use distilled water or isopropyl alcohol. They may take a little longer. It will help to warm them a little, but nothing close to boiling temperatures.
Commercial printhead cleaner is sold in small bottles, often in the ink section of your office supply or computer store. Many of the online companies selling ink refill kits also sell this. Here's an example sold on Amazon: https://www.amazon.com/Printhead-Cleaner-Epson-Printers-150ml/dp/B004VMWTSO (I've never tried this particular product, this is just to illustrate what I'm referring to).
Procedure if your printhead is a user-replaceable part
If the printhead is a user-replaceable part, the printer may even have been shipped with that as a component that you insert yourself during setup. Insertion and removal is usually a simple operation; the printhead drops into the carriage and a lever holds it in place. In this case, cleaning is pretty simple.
- Remove the ink cartridges (temporarily reseal the cartridge's air vent and ink outlet).
- Remove the printhead.
- Soak the printhead in ink solvent in a small container, nozzles down. It wouldn't hurt to elevate the printhead slightly so the nozzles aren't sealed against the bottom of the container. You don't need to submerge the entire assembly, but deep enough so that it will seep into the inside of the head and the ink feed cavity. You can also add a small amount of solvent to the ink inlets where the ink tanks feed.
- Soak for 15 minutes to half an hour. If you happen to own an ultrasonic cleaner, even better; do this in it.
- Residual and redissolved ink may bleed into other color compartments, so:
- Set the printhead, nozzles down, on blotting material, like lint-free paper towels, and let it drain for about 5 minutes. Transfer it to a clean blotter and see if it's still draining. Repeat until it stops draining.
- Let the printhead soak in another container with distilled water, isopropyl alcohol, or cleaner for about 10 minutes.
- Repeat the blotting process, looking for ink. What the blotter absorbs should be either clear or the pure color of the color compartment it is supposed to be. If not, repeat the "rinse and blot cycles", adding a little of the clean solvent to the head's ink inlets to flush those compartments.
- Put the printhead back in the printer, put the cartridges back in, and run a nozzle check. Don't worry if colors initially look a little weak; that's just dilution from residual cleaner that will quickly clear up with a little printing. The important thing is that the nozzle print pattern is good.
- If that doesn't completely clear the clog, repeat the procedure and do the soak for longer.
Procedure if your printhead is NOT a user-replaceable part
If printhead removal is not a trivial process, you can accomplish the cleaning in the printer with a different procedure. I'll describe the steps here, but it's always clearer if you see it in action. I found a video from a printhead cleaner manufacturer that demonstrates a slight variation of the procedure using tools they provide. The video is pushing a particular commercial product. I've never used this one and can't comment on it, but use this basic procedure with that or another appropriate solvent: https://www.youtube.com/watch?v=i78Tu3RVO3s.
- Remove the ink cartridges and temporarily reseal their air vent and ink outlet.
- Even though only one color is clogged on yours, it will be much easier to access things if you remove all of the cartridges. It's up to you whether you clean just the one color, or treat all of them while everything is open. There's a philosophy, "If it ain't broke, don't fix it." On the other hand, this procedure can take awhile, and residual ink of other colors may thicken or dry. So I typically at least add a little solvent to the others.
- Shut off the printer and remove the power cord. This should let you slide the carriage. Move it to an accessible area.
- Fold some lint-free paper towels and slide them under the carriage to catch drips.
- Now you have to get about 2 ml of the solvent into the inlet hole where the ink tank drains.
Use an eyedropper or syringe (a few drops at a time).
Note that the video shows use of a syringe with a tube that is supplied with that product. If you use that product, don't use a lot of pressure on the syringe plunger; this works by dissolving the ink, not blasting it with pressure. If you use too much pressure, you could damage the head, or cause the tube to pop off and spray solvent everywhere.
Especially for a situation where only a few of the nozzles are clogged, you may get better results putting in a few drops at a time, letting it sit a few minutes, and repeating, rather than squirting in 2 ml all at once. In the latter case, the cleaner may drain through the good nozzles before it has much time to work on the clogged ones.
- If the printhead is seriously clogged, you may not be able get all of the solvent in at once, anyway. In this case, if you're using the "syringe and tube method" instead of the "drip method", put in a small amount, wait half an hour, then add more. Repeat until you've put in about 2 ml.
- Once you've gotten 2 ml into each ink port, let it soak for 15 minutes to half an hour (or the time recommended by the cleaner manufacturer). If you go longer than 15 minutes, add a little more cleaner at that point.
- The excess cleaner will drain out into the paper towel. When the soak time is up, slide the carriage back to its parking station and remove the paper towel.
- Put the ink tanks back in, power up the printer, and do a nozzle print test.
- Any residual cleaner may dilute the ink and make the colors look weak until you print a little. Don't worry about that. Just look at the nozzle test pattern to verify that all of the nozzles are working.
- If there is still a clog, repeat the process.
If these procedures don't work
If you can't clean the printhead by one of these methods, you need a new printhead or a new printer.