will GTX960 work on HP 3048h ?
yeah, good chance it would work. However, If you want to play ps3 or ps4 games - I suggest to instead finding a cheap refurbished Dell with at least an i5-2400 or i5-2500k to pair with the 960.
if you have no budget to buy another pc.. then please be aware on the limitations of an lga 775 pc. I have a GTX 1650 installed to an lga 775, so have an idea of what would happen when you install a 960.
1. Attempting to multitask can cause severe lag. For example, you're gaming.. then you alt + tab out of the game to watch a youtube video.. prepare for lag or temporary freeze. I avoid this by simply doing one thing at a time, if you're gaming - close the game first if you want to watch a video or browse the internet.
2. Assuming you have a quad core cpu (anything less, then it's hopeless) and 4GB of physical ram installed (anything less, then it's hopeless). Depending on the game, you can play at 720p or less. And on certain games it just takes some adjustments in graphics options.. for example.. I can play Genshin impact at 60 fps, high graphics settings with no lag - just had to turn off the in game V-Sync.