In the sentence "The person does their job they have got and reports when they have finished it", they and their are meant to refer to the person. Isn't it too much switching between singular and plural?
1 Answer
It is a "singular they", rather than a non-binary they, and has been used for for a long time. There isn't any real problem in switching back and forward between singular nouns and the (grammatically) plural pronoun. There's more problem than would occur with switching with the singular and the (grammatically) plural "you".
The use of "their job they have got" is not great. Either "The person does the job they have got" or "The person does their job". It doesn't work well to mix those sentences.
Other than that, the agreement in your sentence is correct. "Reports" agrees with the singular "person", and "have" agrees with the (grammatically) plural "they".