Student or Learner
"I don't have smokes/money on me."
"I don't have smokes/money with me."(Incorrect, right?)
If you're saying you forgot to take something with you, you can say 'with' in some contexts: 'I'm sorry officer, I was just nipping to the shop for a paper, so I don't have my licence with me'. (Here too, though, you could use 'on'.)