Unfortunately, this particular PSU in your HP PC is propriety. The P1 is used to power up the mobo.
Standard PSU bought outside uses 20/24 pin design.

You will need to get back the same PSU from HP.

I wonder how did u managed to slot in the MSI ? It is Micro Tower no?
Dont think got higher spec psu. u might try shops at low yat plaza.

yea so how did u managed to slot in the graphic card?

Now I'm confused... yu know that it'll not fit but yet you're upgrading to power the GPU?

Ghetto mounting the hardware? well good luck... Sharing the power from a PSU is no issue if there is enough power, but a 350W OEM PSU  for a Gtx650Ti is hardly what I call enough

another option is to determine the wattage those proprietary connectors carry and solder them on a new PSU... if thats even possible

That's a custom/proprietary PSU (possibly OEM is Delta from the label layout), and a standard off-the-shelf ATX PSU will not fit at all. I would suggest dumping the rig, then get new motherboard, new casing and your desired PSU. You can move the processor, RAM, HDD and CD/DVD drive to the new machine...

Do you realize that this proprietary PSU only has 12V outputs (including the standby power)? Can see that from the ratings on the label. I don't think any amount of simple rewiring and soldering may get any off-the-shelf standard ATX PSU to work on your machine (can compare the power supply connections and you will see they are vastly different, including the standby power side). If you want to get a off-the-shelf standard ATX PSU to directly replace that proprietary PSU then some electronics knowledge is required (in other words more complex modifications, especially that standby power section). You could use off-the-shelf standard ATX PSU as a secondary PSU though (using a relay to activate power on/off control), though that PSU will be dangling outside the casing. Depending on the PSU design, some may require a minimum load to get the PSU kept switched on as well. The common ground on both PSUs also has to connected.