Different contracts with the same employer

I agree, Tim Sherry's answer is correct. The sentence is asking "have you had multiple periods of employment WITH THIS EMPLOYER". The phrase "with this employer" explicitly excludes jobs with other employers.

Tim Sherry's answer is correct. Multiple periods = multiple times + this employer. Have you worked multiple times with this employer? Have you worked multiple periods with this employer? I used to work as a temporary attorney in NYC. I would have a project with an employer and then the project would end, and then a few weeks later there'd be another project. So I had multiple periods of employment with that employer.

@tim. I don't think so. That question is asking do you have another employment during last year. Is my understanding right?

This would mean that you spent different 'periods' of time working for the same employer. Example - I worked at the school between 2011-13, 2015-2017 and then again in 2019 for six months. So I had multiple (three) periods of employment with the school.