Engine backfire

Backfires where? Out the tailpipe or back out the intake manifold?

So this is when the backfire occurs, when giving it full throttle?
First of all, retrieve the trouble codes from the ECM. On a 1987 you do not need a scan tool or code reader, just a paper clip and touch together the top row two terminals to the far right then turn the key to run, engine off and count the check engine light flashes. It will flash 12, three times, flash any codes three times each, then return the flash 12, three times when done. For instance a 12 is a flash pause and then flash, flash meaning a 1 and 2 or 12.
Then check the fuel pressure and try some new spark plugs, check the cap and rotor, ignition timing.

Which engine is in this?
Even though you have replaced the fuel pump, have you tested the fuel pressure? Because the fuel pressure regulator could be a concern.
Does the engine misfire at idle?
Did any of the spark plugs you removed show signs of misfiring, where any smelling like gasoline or any covered in motor oil and if so which ones?
Does the tailpipe blow out any smoke, light blue, black or white?
Have you checked the ignition timing?
Is the check engine light coming on when the engine backfires or does it stay on?
Have you retrieved any codes and if so what were they?
Does the tank have fresh gasoline and if not is the gas older than say a year?

This engine is a 7.4l.
What about all of those other questions I asked?

Does this actually have a carburetor, because I see it could have either that or a fuel injected throttle body?
Anyhow, if it has a carburetor, the fuel pressure should be about 4-7 psi.
For throttle body fuel injection, 9-13 psi.

Do a cylinder balance test and find out which cylinder is misfiring. If it does pop and misfire at idle, you can do the test then. The easiest way is to wear some thick gloves so you do not get shocked and pull the spark plug wires off at the distributor cap and see if the engine rpm drops a little after pulling each spark plug wire, then put it back on and move onto the next until you check all eight.
Once you find out which cylinder is misfiring, try a new spark plug or make sure the gap is correct, because sometimes spark plugs come not gaped correctly or dropped and dropping on the end may close the gap.
Make sure you have a good spark and the end of the spark plug wire on the cylinder that had to drop in rpm when removing the spark plug wire from the distributor cap.
If the engine does have a misfire and it's not due to a spark plug or wire, then it may be due to something else, such as a valve not seating or a worn down camshaft lobe. Either of these two can cause a misfire at idle and while revved up.
Even before all of this, double check that the spark plug wires are not crossed, so check the firing order. A lot of times five and seven get mixed up causing a misfire on five and/or seven.

Ken says to check this out, https://www.2carpros.com/articles/engine-backfires-while-running