Hi, Charlotte, welcome to the forum. There's lots of information here. Any company that sends you money, asks that you deposit it and send cash is definitely a SCAM.
There's a great list of about 200 legitimate companies, Official List of Mystery Shopping Companies, at the blue link at the bottom of the page. Each company pays differently, according to its own schedule, but they DO NOT pay in advance and they will not send you checks to deposit nor will they ask you to send money anywhere.
Mystery shopping also does not pay nearly as much as the scammers pretend that they will pay. I think a lot of people get caught up in these scams because they have a check for a bunch of money and the check looks so real ..... and it's so much money they hate to pass it up. Mystery shopping typically pays $10-15 for an assignment, not a couple of hundred dollars.
Being a mystery shopper means running your own small business and working as an independent contractor for multiple mystery shopping companies. It is very time-intensive at first when you are researching to see how mystery shopping works, and you are planning your business and making applications with all the companies.
Good luck and happy shopping!