A CPS(cam position sensor) is used by the computer(PCM) to fine tune sequential injector timing with intake valve opening, and also to some extent spark advance.
If PCM is expecting a CPS signal then yes your "truck" needs that connected, but no "engine" needs a CPS.
Distributorless systems use a CKP(crank position) sensor to set base spark timing and injector timing, and can run fine with good MPG using just that sensor, the CPS was added for better MPG and lower emissions.
If you are not getting a CPS trouble code then your PCM may not be programmed for it, but unless the engine has been swapped that would be an oddball for sure, and not sure why someone would reinstall the CPS on an engine swap in any case, unless they didn't have the oil pump drive from the old 4.0l that came out on the swap.
OR........your PCM is going bad.
But I doubt the CPS or no CPS would account for the very poor MPG, the 4.0l OHV was not known for great MPG in a Ranger(3,000lbs brick with wheels), but 14-16MPG would be more in the ballpark.
PCMs tend to run an engine rich when there is a problem, that is the fail safe direction since Lean causes pinging and possible engine damage.
MAF(mass air flow) sensor is the big kahuna sensor for air/fuel mix, without proper air flow data the PCM is just guessing based on tables it has in memory.
This article has some good info on cleaning and testing MAF:
http://www.therangerstation.com/tech_library/CleanMAF.shtml
MAF sensor problem WILL cause poor MPG and bucking.
Running rich can damage O2 sensors over time, talking months not days, but that should also set a code if O2 sensors are not responding to changes made by PCM.
So another oddity pointing at PCM issue.
Not sure what the swap to manual trans does to PCM overall, I wouldn't think it could effect MPG that much, but don't know.
You might check the resistance or voltage on the ECT(engine coolant temp) sensor, this is a TWO wire sensor just above the t-stat housing on the 4.0l OHV, there is also a ONE wire sender near by for the dash board temp gauge.
The TWO wire ECT tells the PCM engine temp so can be associated with an engine running better cold than warm or visa versa, this sensor rarely fails, but not never fails.
PCM sends ECT 5volts on one wire and gets back approx 3volts on the other wire if engine is cold, when engine is warmed up(200degF) that 3volts will have dropped to under 1volt.
So colder = higher return voltage, warmer lower return voltage, so if testing with an OHM meter resistance will go up as ECT warms up.
For your rich condition this would be a long shot, engine would run better cold which requires richer mix and then run worse warned up.
ECT working will also cause PCM to set high idle when engine is cold, 1,000-1,200rpm, and then PCM will gradually reduce idle to warm idle of 750-800rpm(automatic), manual would be 650-700rpm
If your engine is doing this then ECT sensor is probably OK