You'll probably make it, but you might not. And you might need to run.
Although it is five minutes to change terminals, you will need to clear security at Manchester, which can take a few minutes. If you're running late, make that clear when you get to the queue to the airport staff.
Fifty minutes is a bit on the short side, so you if you are the kind of person who gets very nervous about missing a flight, don't take it.
If you are relaxed about the possibility of being rerouted, and it is a good deal, then I would say take it. If you don't, you'll still get to New York, probably the same day.
Have a back up plan ready if this goes wrong. Don't rely on the airline staff to help you, because they are not very inventive, they might say "wait until tomorrow". In this case, ask them to reroute you via LHR. There's a BA shuttle flight almost every hour down to LHR from MAN and from there a huge number of AA and BA services over to JFK. Since Finnair, BA and AA (and Iberia) are in a joint arrangement for transatlantic flights, there will be no problem re-routing you this way. [This suggestion assumes that you don't need a visa to enter the UK. If you do, this would not be a good option because MAN-LHR is a domestic flight.]
Both flights are with Finnair.
In fact your flight from Manchester to New York is a codeshare operated by American Airlines.