The engine hesitates to pull off at a stop, it makes a stuttering sort of vibration. Could it be the transmission.

Coil on plug ignition system , coil misfiring ? What engine does it have ?

just by your description (detailed) id say it could go a few directions ??4 a cause, one way is the 40 mph is near the speed that overdrive is engaged (42 mph) these do sometimes have problems inside them (gears ) also the stutter you mention is sort of an indication that your torque converter didn't lock up or is locking/.unlocking this is an indication of a immanent torque converter failure , if car was made past 2001 then chances are good it has some kind of transmission control system that could log a fault to help you look for a cause like a faulty speed sensor or something , ck your codes !!

P1336--CKP or CMP Input Signal to PCM Concerns

Check Crank & Cam Sensors for their installation & wire
connector

After that a repair shop will be needed

The code is your symptoms to a Tee

Hello
The Engine and Automatic Transmission (not applicable to manual transmissions) in this vehicles drive train are fully electronically controlled by a computer called the PCM and TCM (Power Train Control Module, Transmission Control Module). When a problem like this or other drive-ability related problems occurs the computer stores a record of the problem (there are of course some exceptions to this, like the fuel pump, engine coolant temperature sensor and MAF sensor for instance) in the form of a fault code in its memory, to read these fault codes you must have the systems memory scanned with a special tool. Once the fault code(s) are read you then must perform the appropriate diagnostic testing to find and resolve the problem(s) DO NOT REPLACE ANY PARTS UNTIL A TRAINED TECHNICAIN HAS DIAGNOSED THE PROBLEM TO AVOID SPENDING YOUR HARD EARNED MONEY ON PARTS THAT MAY NOT CORRECT THE PROBLEM. Also always check fuel pressure for correct spec for your make and engine type.

Sounds like a shorted or sulfated battery to me.

Well I am not a mechanic, but I am fairly handy with a wrench. My initial suspect would be some sort of transmission issue whether it be mechanical or sensor related.Is it more noticeable when accelerating slowly vs. a quick acceleration? That could mean a torque converter. When is the last time that you had the transmission fluid changed? A good shop should be able to rule out many issues by looking at the fluid that comes out (looking for metal shavings and the like). Taking your car to a transmission specialist may be your next viable option. Hope this helps

its the drive belt tensioner dampner it goes bad and sounds like a bad engine vibration when accel and on cold start up

Check to make sure both headlights are tight,,,when it rattles at idle go feel the headlight and hold it and see if the noise and vibration goes away,sometimes the headlight has too much clearance and causes a massive vibration,,if thats not it try...loosening all the motor mounts and rock the car back and forth and them retighten them,it reseetles the mounts and sometimes fixes the issue,,also check the lower engine and trans mounts for wear.

the problem could be related to the tune up of the engine.. when the vehicle is traveling uphill there is more load on the engine, so if the engine spark plugs arent up to par they will missfire causing the hesitation your feeling .. damp conditions could also contribute the the misfire condition .. i recommend having the tune up parts checked and replaced as necessary.

Sounds like you may have a worn out CV joint and the clicking may stop with an oil change. Good Luck