The brand doesn't really make a difference in this situation. You are running on minimum power. HP uses PSUs to power the stock build adding any extra hardware and especially a GPU will probably call for a PSU upgrade.
You could always give it a try but I would suggest picking up a new PSU.
Depends on what hardware you have in your system, a 6670 can only use 75w max due to not having an external PCIE connector:
Though, I suggest getting a 7750, it uses the same power and is around the performance of a 6770
My personal opinion on this is, it might work its just a question of how long.
Maybe take a pic of the side of the psu label so we can see how much power it can really output? Most OEM (like dells, gateways, etc power supplies) manufacturers lie on the labels and put down whatever they feel like.