Know that the belt can look ok on these and still slip. You could try also just creating a bit more tension on the cable temporarily to see if it improves. The trans RNR means it all comes apart. The wheels , arms, housing baffle etc If you go that route, it would be best to have your new trans sitting there and ready to go in. You would need to post the complete model / serial to know for sure- (or maybe check plano power) as generally the transmissions do not have servcie parts available for these.

Here's an informative thread on the HRT216:

http://forums2.gardenweb.com/forums/load/lmower/msg1214544622141.html?1

I need to replace the left handle backet on my Honda HRT216SDA mower. Do I need to remove the entire drive system to get the 3 bolts that attach this bracket to the mower deck?

HRT 216. There should be more numbers after it.
Anyway, the 3 speed transmission is a known problem. In most cases it needs to be replaced.
I have mentioned before that the trans leaks oil and the plastic inside gets destroyed.
I have drilled a hole in the bottom and threaded for 1/4 20 refilled the oil and gave them a new life.
There is no pressure relief so when it gets used a lot it leaks.