Can/does Turbotax file CCA tax for Ohio cities?

Yes, TT can prepare a CCA form. But, it  cannot e-file an Ohio City, CCA  or RITA return. It must be printed and mailed.

TurboTax (TT) does not automatically prepare a city return, you must initiate it. At the screen, in the Ohio program, titled 
 "A few things before we wrap up your state taxes"
      scroll down to Ohio City taxes. 
Or, later
"Your 2016 Ohio Taxes are ready for us to check"
    -Click Other situations
    -Scroll down to Ohio City taxes. 

I actually find it easier to fill out a city return by hand or at the city, CCA or RITA web site, than to use TT. Things don't happen automatically and you need to know what you’re doing. The actual  RITA/city forms are easier to follow than the TT interview.

http://www.ccatax.ci.cleveland.oh.us/