Likely a head gasket problem, between the water jacket and the one cylinder. Look for bubbles in the radiator with the radiator cap off before it heats up. Or run compression check on that cylinder. See if it holds the pressure after you stop turning over the engine.
Error code p2101 involves the electronic throttle control. Which essential controls your throttle, the car won't accelerate if the code went away and came back after you changed the battery chances are it's just the throttle control sensor which isn't that expensive to replace.
What size engine ? The clicking you hear is probably the starter relay , provided you hear the click while turning the key to the start position . That black box is the fuse / relay junction box. Testing the starter electrical circuits with a volt meter is the way to find the problem . Voltage drop testing . Free wiring diagrams here http://www.bbbind.com/free_tsb.html Enter vehicle info. Year , make , model and engine size . Under system click on engine then under subsystem click on starting . Click the search button then the blue link.Does your vehicle have a manual or automatic transmission ? First blue link for a manual transmission ,the second link for a automatic transmission . How to voltage drop test starter motor circuit
YES you need them, ENG MTG is the Engine Management system, and the ENG CONT is the
Engine Control system so you see if they are disconnected you will have a good time wondering what is up with the car
If you have done all the usual tune up stuff, check to see if holding the gas down a little helps. If so, could be a dirty/sticking idle air control valve. Other possibility is bad coolant temp sensor telling the ECU engine is warm when it is not.