Can't bootload ATMEGA328P-PU

[system]

Assuming you removed the connection from AREF there is just one obvious wiring deficiency. You really should have two 0.1 µF capacitors connected from VCC to GND as close as possible to the target. I've programmed targets without bypass capacitors so that is probably not the problem.

Given what you've posted there are just three possibilities left...

1. Broken or loose wire.
2. Clocking problem.
3. Programmer's bitrate.

Only you can do anything about #1.

#2 seems unlikely. The capacitors are in range (12 to 22 picofarads). Everything appears to be connect correctly. There are two clocking options in boards.txt (internal oscillator and external crystal / resonator) so it is unlikely the fuses have been set to something "odd".

Which leaves #3. Your programmer has to be configured for a bitrate of 125K or slower. The avrdude command-line you posted does not include the option to set the bitrate which means it could be anything. I assume your AVRISP MKII has a way to set the default bitrate. I suggest ensuring it is set to a reasonable value.

@Coding Badly thank you for spending so much time on this.

I removed the Aref connection per your earlier post and it didn't solve the problem... but I left it unconnected
I added supply bypass caps close to the micro which also didn't solve it, but I left them in as they should have been there in the first place.

I doubted it was point #1 above as I completely rewired the system on a different breadboard with new wires.

Your point #3 gave me an idea. I hooked up the AVRISP MKII to AtmelStudio. Its clock was set to 2MHz (less than 1/4 of the 10MHz). I tried reading the part and it didn't work. I tried reducing the clock to 1MHz and it still didn't read the device. However, I set the clock to 125kHz and it DID read the device. I'll see if I can program the part now.....