My car has a little miss and the idle goes down then goes up when I first start

What is your question?

My battery died a while back, ever since I have problems with the way it runs. The computer was reset when the battery was jumped.at first it was the check engine light then the wrench. I went to my mechanic he set the computer codes that were off it runs good for a while then I go start it and it runs likes I said above.my husband thinks its the coils and plugs this car has 44.700 miles on it.I think its something to do with emissions.