Sorry to tell you this, but Mom's PC has reached the end of its useful life for some reason. Those are 'blown' capacitors which are quite important for the system to work, and although they CAN be replaced at an electronics repair shop I'd suggest that the exercise would be more expensive than the PC is worth.
The best way to proceed would be to get a new 'system box' (which is basically a new PC except for screen, keyboard, mouse etc. Unfortunately, The version of Windows and installed software which came with that Pavilion can't be reused on a different system, so you'd need new copies also.
If Mom has important files on that Pavilion it's a rather easy job to remove the hard drive from the old system, bung it in a new system as a 'secondary' drive and copy the data files across to retrieve them.
Sorry to be the bearer of bad news, but 'leaking' mainboard capacitors are a sure sign of imminent death!