Ask any Englishman what ‚half time‘ is and they will tell you that it’s the break in the middle of a football match. So a ‚half-time job‘ would be something like a person who comes on to the pitch and repairs the cut-up grass, or sprays some water. Or a man in a bear costume who entertains the crowd.
If you are not talking about these jobs, but instead, you are talking about a job that you do not do permanently, you need to use the phrase ‚part-time‘.
So if your cousin works a few hours or a couple of days in a supermarket, he has a part-time job.
We also use the phrase jokily for people who we think do not do much work. He leaves work on Friday at 2? What a part timer!