Learn Polish: On chce obiecać klientce dobrą cenę. - He wants to promise the female customer a good price.

This is the dative case, which is used for the indirect object (the person receiving the action). He is promising a good price to the female customer. For feminine nouns ending in -ka, like the base word klientka, the ending changes to -ce in the dative.

Why does the letter k change to c in klientce?

When we add the dative ending -e to feminine nouns, the consonant right before it often has to soften. In Polish, the letter k regularly changes to c before this -e. You will see this pattern in other words too, like matka (mother) becoming matce.

Why do we say dobrą cenę instead of dobra cena?

The phrase dobrą cenę (a good price) is the direct object of the verb obiecać (it is what is being promised), so it takes the accusative case. For feminine nouns, the standard -a ending changes to -ę, and the matching adjective ending changes to -ą.

Is obiecać a perfective or imperfective verb?

Obiecać is a perfective verb, meaning it focuses on the completed result of making a promise. If he were making promises repeatedly or generally, you would use the imperfective form, which is obiecywać.