i have almost same hp 600 g1 sff with gtx 750 in it. its been working fine since last year.
you have to buy a low profile card with max 75w power consumption to fit in the cabinet.
also its better to keep the side panel open when gaming or heavy gpu tasks.
post power supply pics if possible.
mine has 80+ platinum made by Delta so i was confident that gpu will work fine. i would not have installed the gpu if power supply was made by some other company.
that will work but
1)any card will take 75w from onboard pcie. anything more than 75w has to come from pcie power connectors.
2)the cards that can directly fit in SFF case are only low profile cards
3)all low profile cards consume less than or equal to 75w
so extra psu is pointless unless you want to put a card like gtx 1060 or rx 470 or more powerful ones.
post a picture of psu .. we can determine the quality of psu by manufacturer