First, if you are a couple (or a family) and one member is inadmissible, all members are similarly inadmissible. So they reject the entire family/couple, not just one individual.
A Temporary Resident Permit overcomes a finding of inadmissibility and permits the inadmissible person (or people) to enter Canada. It does not authorize work or study - you still must have a work permit or a study permit - but it allows you to come to Canada despite your inadmissibility.
OP 20 "Temporary Resident Permits" describes the CIC procedure for evaluating such applications. You can find it on the CIC website. If you wish to pursue a TRP I strongly advise working with an experienced immigration attorney because such an application has to be persuasive. In the case of a PR application, what you're basically saying is "if you don't believe me when I say I won't use these social services, please grant me a TRP so I can demonstrate my willingness to do so."
For example, some things are difficult to do when you are not in Canada - like find an employer with group health insurance coverage for prescription medications. Thus, being allowed into Canada for a period of time to find work (and then get a work permit) is one way to demonstrate that you are acting in good faith. I have not seen a lot of arguments in this area, but I do know that they can be made.
Fair warning: I actually applied for a TRP in February 2012. The visa officer did not even start reviewing it until September 2012, at which point she learned my second PR application had been approved and thus decided not to continue with it. She did tell me that she was inclined to grant it, but ultimately it was not necessary. When I was doing my homework, I found that CIC grants fewer than 200 TRPs for medical inadmissibility in a given year so keep in mind this is a long shot approach. That's why having someone working with you that has done this successfully is an excellent idea and one that improves your odds of success.
TRPs are discretionary. CIC does not need to grant you a TRP - you have to convince the officer that it is reasonable under the circumstances. Cases where applications are granted generally involve H&C considerations, early admission spouse/partner applications, and national interest cases (e.g., if you're Stephen Hawking and you want to come to Canada to work or live, you ask for national interest consideration because you are a unique individual with considerable prestige to offer).
Thus, you should view the TRP request as the last ditch effort in a well prepared application. Assuming the medical condition is valid, you can discuss costs (especially when CIC doesn't show costs) and explain how your treatment doesn't exceed the excessive demand threshold (assuming you have a treatment plan that doesn't exceed it). Then you can discuss ways in which you will mitigate those expenses - set up an irrevocable trust account to pay the expenses, arrange with an NGO to supply the medications without cost to the Canadian government, point out that the costs are not uniform because some of the drugs will fall out of patent status soon (e.g., in the case of HIV, the components of Atripla - the most commonly prescribed treatment - are losing patent status in the next few years and that will drive the cost of the therapy down. You don't have to take the "convenient" one pill formulation, which is more expensive because the combination is patented, but can instead take two or three pills with each of the individual components that are off patent). But it takes a lot of research to find the patent expiration dates for these drugs. Since CIC does not usually report pricing I have used pricing information available from Canadian pharmacies that specialize in providing these drugs (often using foreign produced generics). If they don't provide pricing and you DO provide pricing - from a Canadian pharmacy no less - it is difficult for them to rebut you. I've also used the Quebec price list (published by the Regie) which is representative of the price for brand name or legally available generics in Canada.
A good response to a fairness letter can be quite lengthy, especially if you attack on multiple grounds. Part of the strategy is to lay out arguments that can then be used should you pursue it into court.
The TRP is there for the "just in case you don't accept any of the previous arguments..."