That's your connection to the reset board... For the most part unnecessary once you get everything up and running properly.
White with orange stripe (plug pin 6) should go to pin 24 of A24-P2 (cathode end of diode on DIP adapter). Pin 24 of TC1 is actually an unused position and connection to the CPU board is actually made through that diode. This is the reset output from that reset board - being disconnected shows the reset board has been doing -nothing-.

Is that yellow wire really tied to pin 17?!?! Pin 17 is your CPU reset line and this connection would render the entire reset module useless. That should be tied to the IRQ line which is on pin 16.

On that machine, the function of this reset module is to monitor the CPU board's IRQ line. If it goes inactive (stuck high or low due to locked software and/or hardware), the reset module would time-out after a brief period and send a reset pulse to the CPU board to restart things (hopefully).

And - if the caps on the reset board are originals... they are small value caps which tend to go bad as time progresses. If you plan on keeping the functionality of the reset boards, I would replace caps C1 thru C4.

Ick. That board has quite a few hacks on it. The red wire going to the 6502 (D1 from the looks of it), the green wire to the via on the board and I think A0? at the pinout on the daughterboard. Also looks like some hot glue under the daughterboard / eprom board so that may have been pulled and "fixed" at some point since that daughterboard is notorious for cold/cracked solder joints. I'd guess that's part of what's causing your board not to boot and it's the first thing I'd clean up if I was repairing the board.

I'd have pointed you to this.. http://www.pinitech.com/products/gottlieb_piggydeux.php -- however, considering some of the hacks I'm seeing I'd have to see the back of the MPU too to know what kind of other hacks or pulled pads/traces might be on the back. Sorry if any of the hacks were your attempts to fix anything, I'm just pointing out what's visible

I've also got this available, that might help you diagnose the board further depending on your skill level:
http://www.pinitech.com/products/gottlieb_universaltester.php

But it may also be more cost-effective or save you time to just send the board out for repair to someone like Chris Hibler @ http://chrishiblerpinball.com/ Just depends on how far you're wanting/willing to go yourself in regard to money and time spent.. or if you're set on learning & fixing yourself.

This was attached incorrectly but won't be the source of your problem. Just leave that plug uninstalled to the reset board until you figure out why the main CPU board isn't running. Disconnect the 7 pin plug from the reset board. Leave that 40 pin DIP adapter installed as that 3.3K resistor on the adapter is a pullup for your Read/Write signal. This resistor greatly helps with some of the more stubborn boards.

Your Hollywood Heat CPU board has a Genesis Game Rom (705). Hollywood Heat should be 703. I agree with acebouthound: replace the piggyback and clean up the mess around it; by a pro if you are not experienced with PCB repairing.

Marco

The jumpers appear to be there due to broken traces often caused by futzing around with that EPROM adapter. The jumpers on your board go between the U3 (where adapter is plugged into) position and components to the left of U2 (beneath adapter) position so there is probably a trace break between the U2 and U3 positions. Most probably at U3 where headers are soldered.
The plus side to using a replacement adapter type that plugs into new U2/U3 IC sockets is it will actually jumper over breaks between U2 and U3 and also provides redundant connections to signals that don't have breaks.
https://www.greatplainselectronics.com/proddetail.asp?prod=140-101
This does require you to install normal IC sockets at U2 and U3 positions whereas the other adapter just plugs into TC1.