1995 Ford F-150. Engine Size : 5.0 V8.
Question: cuts out and or backfires through the intake at different speeds coasting or pushing the pedal to the floor corrects for a short time.
Answer: Well, cutting out and backfiring are two totally different things. If the engine is just basically running bad and backfiring through the intake manifold, this is either an ignition problem like arching spark plug wires, cap and rotor or bad spark plugs, or a valve-train problem. Worn cam shaft lobes or valve spring problems will usually cause a backfire through the intake. But coasting or pushing the gas to the floor to correct it would not make sensor for any of the above.
Ignition problems get worse during acceleration and valvetrain problems would do it all the time, no mater what. You could have a sticking EGR valve, oil or coolant fouling a spark plug, a bad throttle sensor or any number of things. Going to have to get this checked out by a mechanic.