You aren't going to be able to find out if the card is functioning properly if you are never supplying it with the required power. The results of underpowering a card can be artifacts, no display, visuals like you are showing, and sometimes even appearing to work fine until you put a load on the card where you get crashes. etc. But if there is a problem with the card, you'll never know it if you can't meet the requirements. And by "you'll never know" I mean the card will always appear to have issues while your power supply is too small.
WILL upgrading your power supply fix the issue? No Idea. In fact, no one out here can know if it WILL fix your issue. But, until you get a 400 watt power supply, you will never know if it is the cause or not. You will need to upgrade your power supply to eliminate it as a suspect.
EDIT after seeing your comment about having a Dell Studio Slim (540s) you will need to look for a TFX power supply for that unit. Something like this one, or this one. I actually pulled the link for the first one out of the Dell Support Forums.