2010 Mazda Mazda3 smoky tailpipe with jerk during shift

Hi, So my car didn’t start so I had it jumped.I was thinking dead battery. It started after jump but now there’s smoke from the tailpipe and it jerks when I shift from park to drive. Smoke is white/gray color. Thanks.

Could be the alternator isn’t working right.
Check battery (system) voltage with engine idling.
Should be ~13.5-14V.
Could also be an electrical surge during the jump damaged electronics.