I don't know. It sounds like their MP is either incredibly naive, trying to take credit for something that will happen anyways, or corrupt. Let me explain:
Once you send in your Notice of Appeal, you are pretty much guaranteed a hearing, unless it gets cleared up at ADR. So saying that they will try to get you a hearing if you file the appeal is like promising to try and make sure the sun will come up tomorrow, just for you.
If any MP is able to get someone's case to the front of the line, then every MP would be besieged with such requests, pretty much nullifying any advantage, as someone else will surely be bumped ahead of you. If a singlular MP did have such influence, and were to use said influence to help someone jump the queue, ahead of all the other people who have been patiently waiting their turn, then that would be influence peddling, and my definition of corrupt. Any system that would allow that would be a corrupt system, IMHO.
Imagine if you have been patiently waiting for your hearing, not knowing that it has been bumped time and time again by the actions of a certain MP. Especially if you are in one of the jurisdictions where immigration hearings are only held six times a year, or less.
Also, as noted above, MPs seem to be hamstrung just making inquiries on your behalf, so I wouldn't expect much in the way of anything beyond that:
http://www.hilltimes.com/2016/08/17/directing-mps-through-new-call-centre-for-immigration-cases-disastrous-say-critics/77540