They would probably only have their main home in London if their constituency was in London. Many MPs have two homes - one in their constituency and another near parliament. So for example they might have their main home in say, Sheffield, and a flat near Westminster. They get an allowance to help them pay for their second home.

There's been a lot of criticism lately of this system as some MPs were accused of abusing this system to get a second home despite living in or near London.

But to answer your question...There are many places where you could get a flat like that. I know that one of the MPs involved in this scandal had a rented flat in Lambeth which is just across the river from Westminster so quite central.