How long a delay are we talking about? My CX-3 basically turns right over within say a second of pressing the start button (with the brake pedal depressed), unless I purposely don't have my foot on the brake which just turns on the power, then I hear that high pitched whine you mention.
When you say choppy do you mean it revs a bit before turning over? On a previous vehicle I owned I had this happen from time to time and was told it was the battery charge level. Not saying this is what may be happening in your case, but it's worth having the dealer check out in any event.
Current Vehicle: 2017 CX-3 Touring AWD in Dynamic Blue
Previous Vehicle: 2015 Nissan Juke Nismo AWD