Unfortunately there's really no way to speed up processing. MPs don't have the power to have your application be prioritized over others so that it's processed faster.
Note on this: I agree an MP cannot directly 'influence' the process or have the power to get an application processed faster.
But an MP
can ask questions of IRCC, and - if the MP thinks it is warranted - ask questions of the Minister's office. Or even raise it in parliament.
And
sometimes those questions do result in files being processed or decisions being made.
And even more rarely, those questions (public or direct to Minister's office, etc) result in some attention being paid by the government (and hecne eventually 'the administration') to issues of policy, including things like prioritization.
I should emphasize:
most inquiries through MP's offices will get nohting more than routine answers, because in fact they are routine. This is especially true of inquiries about applications that are still within the normal timeframe.
That said: it's a democracy, you have the right to ask your MP, and some MP's do actually try and occasionally get results. Expectations should be low: because most inquiries are routine and/or many inquiries are actually about decisions / processes that are handled correctly (albeit not from perspective of applicants). And it does also get MP's to pay
some attention to issues that matter to Canadians.
Also, it is something that can be done and potentially has some impact, as opposed to so many other pointless activities.
But to emphasize again: expectations should indeed be low.