Will treatment for chlamydia also clear up cervicitis?

No it will not. Your doc will probably need to prescribe a cream for the cervicitis.

Yes it should IF the cervicitis is being caused by the chlamydial infection. Otherwise, the doctor needs to figure out the cause of the inflammation (infection, injury or otherwise) and treat it.

Often, chlamydia inflames the cervix (i.e., causes cervicitis), and results in "cervical friability" which means there are sore, soft, red and blood-spotted patches on the cervix.

But killing off the infection will alleviate the inflammation. Just make sure you abstain while taking your meds, make sure your partner gets tested and treated, and both of you complete the treatment course before resuming sexual activity. Also, use condoms to prevent future infections.
I work for your friendly local health department.