CPS or fuel pump gremlins on 99XJ?

I've done enough searching to narrow my choices:
Each morning my 99 XJ w/56K cranks 6-7 secs and stumbles and sputters until I accelerate out of the driveway--then it pulls strong. I've tried the wait 1-2 sec after turning the key on and I've done the key on-off thing.

The stumbling ONLY happens in the morning. After driving I shut off the jeep and it will take another 6-7 sec crankathon but will not stumle upon start-up.

*I hear the fuel pump click on and off after turning on the key.
*my fuel pump is not noisey
*there is no hesitation with 1/4 tank or less and left turns
*I've replaced the fuel pump relay in the fuse block
*my milage is a horrible 9-11mpg all city (30" tires without gears)

I hope it's the CPS=cheaper
It's probably the fuel pump=more $ and a pain
Maybe the MAP?

Either way I'm a good enough wrench to replace either and I've read up on both proceedures.

Thanks

Check your IAC for the startup and your O2 for the poor mileage.

Have you tested the fuel injectors to see if there's one leaking or failing to close?

Aw, shucks. I thought I had it narrowed down so well. Good ideas. I'll have to take it in and pull some codes. Thanks again.

lol.... Yep. There can be lot's of gremlins crawling around. Pulling codes is the right start. (no pun intended).

By all means pull codes, but remember you'll get a code from an electrically defective injector, but not a mechanically defective one. If it's clogged or dribbling, it probably won't show up unless you test the injectors themselves.