Why are you thinking about applying for a TRP? They are specifically for people who have been found inadmissible to Canada, and normally I think they're issued as a result of an inadmissibility finding . . . sometimes by an officer at a Canadian port of entry, and sometimes by officers within Canada when there are compelling reasons for an inadmissible person to stay.
How long it takes to sponsor a dependent child depends on which overseas embassy is processing the application. The
timelines are on this page of the CIC website.