"Minimum power supply is 300W" this line just serves as a guide for customers. In rea llife situations your card uses less than 300Watts, for a low end GPU like the 610, 620, 630 and such, I don't even think they'd use 150W on load.
With regards to your Pre-built PC from HP, the manufacturer of the PSU wasn't mentioned on the link, hence it will be safest to assume that it's a generic PSU which commonly translate that 300W generic PSU will only supply less than 300Watts in real life, plus the fact that you've been using your computer for some time now that PSU must have degraded over time.
If you have the budget and the PSU size fits your case, try this Antec EarthWatts Green EA-380D or SeaSonic SS-300ET Bronze 300W or CORSAIR Builder Series CX430, those may be the best 300-400W PSU you'll be able to get for less than $50.
With those PSU you'll be able to run a 610, 620, 630, 640. Not quite sure but might be able to run a mid range card like the HD 7750, HD 6670. My source was Logical Increments and Techpowerup.