My son has a 1992 Honda Civic and when he first turns the key on if you do not here the electric fuel pump run it will not start. If it does start and you stall it or shut it off you have to wait a minute or two and sometimes longer before it will start again. Sometimes it has been days. Sometimes by pushing it, it may start but not for sure. This is a nice car but a very annoying problem. We are hoping someone can help us out.:)

Replace the fuel filter located:
Under hood, center, upper firewall, mounted in fuel line.
It's just a starting point from the described symptoms.

If the Check Engine Light stays on, when the ignition switch is turned to ON, the ECM is likely failing. Perform the K-Test on the ECM:

The K-Test: Remove the MAP Sensor connector and turn the ignition switch to ON (not start). Using a multimeter, check for 5 volts going between the MAP Sensor connector's reference wire (+) and ground. As you look at the connector, this is the socket on the right. Really press the black test lead into a cleaned main ECM ground on the thermostat housing. If the voltage is low, it's probably indicating ECM failure. Most failed ECMs will record a fraction of a volt. To me, the K-Test is simple, elegant, and accurate.

Here's how to replace the ECM yourself:

https://www.askmehelpdesk.com/cars-trucks/faq-how-troubleshoot-repair-maintain-hondas-selected-other-vehicles-46563-3.html#post235038

The ECM controls grounding to Terminal 8 of the main relay. If the ECM is failing, Terminal 8 will only intermittently ground. When it grounds, the second relay inside the main relay will click and allow the fuel pump to run--otherwise, it won't. It's a complicated "dance" that only takes place when the ECM works. ECMs on these older Hondas are failing frequently.

If the ECM fails the K-Test, remove it before buying a replacement, in order to get an exact part no. match.