"cleaning job" is correct and usable in written English.
You can use it when referring to someone's job or when suggesting an activity. For example, "Jane has a part-time cleaning job at the local hospital."
"The cleaning job," he says, "is still up for grabs".
When Jori's father left her, she had a cleaning job at Mainstay Suites, by the airport.
She's 19 and on her way to the cleaning job which she combines with her studies.
I got a cleaning job at Gatwick – where I had once slept!
He also attributed the poor cleaning job to "poor administration by the county".
Then Moselle, a young mum with two toddlers and a cleaning job, complained about the noise.
Agnès is often offered cleaning jobs outside the cathedral.
People around him started mentioning the places they had worked: factories, restaurants, cabs, cleaning jobs.
Mornings and evenings, Bonnie works two cleaning jobs to cover her rent.
The first time Mr. Pineda did the raffle, there were about 30 names in the hopper for a handful of cleaning jobs lasting one month.
Dorcina and Yonel Noel immigrated to the area from Haiti in 1990, and friends of the family say they initially worked at cleaning jobs at a local hospital.
