Now we're getting somewhere.
The CEL is normally on, but yours clicks and flickers off. Got it.
You have an EEC relay (PCM), which needs current flow through the PCM Power diode (together plus the ground, they light the CEL - BIG clue). Check both of those items' security and integrity in the fusebox (under the hood). You may literally have to dismount the fusebox to access the underside. You may find that the terminal spades have worn out making intermittent contact with the relay spades, or same issue with the PCM Power Diode, simply loose contact.
The final part of that circuit is that when the PCM relay is energized, it routes power to energize the Fuel Pump relay, making it intermittent as well.
Lastly, Ford locates grounds in places you may not look. DS fenderwell, radiator support, etc. Verify the grounds are tight and clean, particularly those I mentioned and any coming off the Bat- to wherever it goes.
My approach would entail a guy in the seat watching for the CEL to flicker while I wiggled wires, relays, diode, grounds (key in Run, engine off). Bet you find it!
Could be as simple as a bad PCM relay itself.