The best place to check would be the specifications on HP's website or your box (if you still have it) or manual (you may not have a printed one, but there is almost certainly a PDF somewhere on the machine or the media that came with it.
If the official specifications state explicitly that SDHC is supported then it almost definitely is, if it doesn't then it almost certainly isn't.
Be careful with 4 GB cards - there was apparently some confusion in the standards around that point and there are some combinations of card and reader that do not like each other.
The "external media" PDF listed at this page for the machine only mentions SD, so you are probably out of luck. The search suggests that there were a couple of revisions of the mode, so you may want to look deeper.
If the machine itself doesn't support SDHC then USB readers for the format are cheap and easy to find. I have a couple of nice mini-USB-adaptors for microSDHC cards for my netbook (they stick out from the port by only about 5 mm) that were very inexpensive, cheap and haven't stopped working yet (these might not be the exact model, but they look very similar).
Unfortunately, the best answer I can give you is that, no matter what the specs say, they may be wrong. Hopefully anything within the last two years should read SDHC.
Your best bet is to find a friend or coworker with an SDHC card to let you try it. If you don't have any friends with an SDHC card (or just don't have any friends...) then you can possibly go to a Best Buy or other retailer where there may be someone working that can help you try it out. I know that after being taken over by Tiger Direct, CompUSA seems to be more helpful if there is one in your area.
Some retailers will even sell you a card, let you try it in your laptop, and immediately return it if it doesn't work. Make sure you ask about this first as not all will do this, and you could be stuck.