In my experience, they don't even enquire themselves. They delegate to one of them collaborators with fancy titles and even then they come hands clean and tell you they can't pressure or do actually anything. What CIC will see as enquiry from the MPs office. If that is reason enough to feel compelled to go into your file and do something with it I think it's also unlikely. I think you got lucky with your interactions with your MP OR your MP has quite a bit of influence on immigration matters
They have zero influence directly on decision, but in the note that showed as MP as done inquiries that date and so on.
Having a call from this line of politicial seems to be more effective on agents.
Maybe was a luck in my case or not .. we will never know but for me it worked.
Asked 3 times for update by them
1st ) got the Mexico Transfer Letter in October 2019 (1 day later)
2nd) got the AOR2 in February 2020 (2 days after)
3rd ) got PPR ( same day in October 2020)
so ... idk what that meant but
in Québec MP refers to " Centre de renseignements des députés et des sénateurs " and seems this line to be more direct to vo's.
The MP even sent me a letter after she was confirmed to be arrived (COPR soft landing) saying that he was glad to been able to help with that case and wishing her Welcome to Canada and Québec.