Can I work in Italy, and does mentioning it hurt my application?

Short answer: Students can work within limits, and the limit is in the rules rather than a secret. But a study visa is granted for study, and your financial proof has to stand without the work.

What the rules allow

A study permit allows part time work within an annual hours limit. The program answers are B37, M43 and P33.

Why the money proof cannot lean on it

The €10,179.85 is proof you can live in Italy from the day you land, before you have found any job (V7). Work you do not have yet is not financial proof, and part time student work does not cover a year of Italian living costs in any case.

On whether to mention it

Your file asks you about your funding and your plan, and you answer those accurately. What I will not do is coach you on what to leave out, because that is the first step toward a file that says something other than the truth (V18). Answer what you are asked, accurately, and no more.

The real trap inside this question

Students who plan to fund a degree by working arrive on a budget that does not exist, and spend the year in trouble. If the money is not there, work is not the bridge. V11 has the honest alternatives.