Give specifics of what you want to do and why.
This is really important, no point trying to figure out what you need.
It may help to know that by custom in Canada, many institutions will just not include your middle name if you ask them (like at banks, etc). In some cases even driver's license or similar. Often people will use a shortened version of their first name, or switch and customarily use their middle name, or initials, or sometimes different names.
So if you just don't want to use your middle name once settled in Canada - just don't use it; that approach will work for most stuff. Then if you really want to get it off your PR card and the official ones, you can change your name legally (whether it works for passport is a question for your home country).
But for the task at hand: best to supply full name as in your travel document - there is a possibility it would get delayed if you don't include. If you want to include a note with your app that you'd prefer First Last and not First Middle Last on travel documents, maybe it will work.
But from what I've seen, IRCC almost always prefers exact match with what is in English in passport (eg if passport has non-latin characters also). [There are some important exceptions like applicants with only a first name, no last name - iRCC makes them use it as last name.]