To counter your intuition, an expensive House race costs about $2 million, whereas a Senate race is closer to $10 million. Maybe a candidate from a wealthier district has $1.5 million more in their campaign chest than one from a poor one. But that's not all that much compared to the $10 million they now need to raise.
Another thing is that not all of your money needs to be raised from your district. You may get cash from a corporation with lots of factories in your state, Emily's List, and so on. Wealth in your district doesn't directly correlate with campaign funds previously raised.
Finally, if you're in the House and running for the Senate then you start state-wide fund-raising far ahead of time. You have a slight edge if you have a good relationship with wealthy home-district donors. But no matter who you are you're mostly asking people in districts you've never represented, and asking statewide organizations (which any Representative, rich or poor district, has an equal shot at impressing). Putting that another way, suppose you're a Democrat from a wealthy city running for the Senate. A Republican House member from a poor district can fund-raise from conservative millionaires in your district. They'll probably give more since they hate you extra.