So we recently bought a 95 Honda Accord LX. The problem with it is that sometimes, when turning the key to “ON” The P light on the dash lights up and the D4 light doesn’t. When this happens, the car cranks normally, but does not start. However, when the car wants, the D4 light pops on and the car cranks and run like normal. Problem too though is that it will run for a good while but even while driving, will turn off again and doesn’t not turn back on with the same issue. “D4” light not lighting up... any suggestion?

 

This almost sounds like an intermittent main relay. It's located up under the dash. Most times they just need to be re-soldered. It's kind of a common problem.
I'd wait until someone with more experience can chime in though. If you have a FSM (Factory Service Manual), they go thru a testing/trouble shooting procedure in it.