I’m having starting issues. Yesterday and this morning it wouldn’t start. I turned the key to on without starting the car and left it for a couple of minutes and tried again. It started up easily after that.

So my question is:
What would cause my car to need the key turned to the on position for several minutes prior to being able to start the engine?

History:
Yesterday, I started the car and it shot a few blanks before it started. I drove it to the gas station and decided to add oil while I was there. It started up normally. I apparently cracked the oil cap when I was tightening it down. I drove it about 50 miles with the broken cap. It had some trouble when it idled and when I was trying to speed up. The car sat for a few hours, then shot a few blanks before it started and ran roughly as I sped up. I parked it again and it wouldn’t start unless I left the key on for a few minutes first. I drove another 50 miles and finally discovered my cap problem. I replaced the cap and read the codes (538, see below). The car started up fine as I left the auto parts store. BUT… I had to leave the key on again this morning to start it. So, something is wrong.

Am I wrong to separate the starting problem from the idling/speeding up issues? I was considering the later a part of the oil cap problem and the starting issue a whole different can of worms. No?

Codes:
538 = "Insufficient RPM change during Dynamic Response Test (Engine Run Self-Test). or,
Invalid cylinder balance test - throttle position movement. or,
Invalid cylinder balance test - cylinder identification problems."

Previously (~Dec) I had the check engine light flash a few times when it was in the negatives. They were all EGR valve problem codes. I haven’t seen them since (we aren’t seeing any negative weather anymore).