Car will not start sometimes unless I plug it in while I turn the key.

Some days my car starts just fine. Other days there will be a click, click when I turn the key before it starts, but it will turn over after a second and run just fine. Other days (seems to be when it's a little chilly out - not cold) there will be no sound whatsoever when I turn the key, but if I immediately plug my car in and turn the key the car will start fine. Last night it was a little chilly so I plugged my car in over night (total overkill because it was above zero Celsius but I wanted to be sure the car would start) this morning I unplugged the car and tried to start it - there was no sound whatsoever again when I turned the key. I plugged the car back in, immediately turned the key and it started just fine. It seems to have more of a problem starting if it gets a little chilly out. It will start fine for days, then one day there will be a couple of clicks (power still on the dash) then it will be fine again. Then one day if chilly no sound whatsoever but ALWAYS if I plug it in and turn the key it starts fine. It has been unusually warm here for more than a week a spring like and it started just fine everyday, several times a day. First chilly day and again it wouldn't start. Do I need new spark plugs maybe? Do I need to clean battery connections? What could it be? Thanks for your help!