Could personal problems of a Customer care officer impact their dealing with customers?

It should not be. Whatever your do the job, you must be attentive perfectly to deliver.

The answer is definitely "yes", personal problems do impact their dealing with customers.

Any state encompasses three elements, thinking, feeling and behaviour. Thinking negatively produces a negative state, feeling negatively produces a negative state, behaving negatively produces a negative state and personal problems may impact the three elements negatively.

Personal problems should not impact dealing with customers, easier said than done. 

Employee should not let personal issues affect their work.