So the reason behind this clutch relay failure is that compressor has become faulty. It would be cooling alright but it is putting load on the engine that is why the relay or clutch got overheat and got burnt.
Same thing happened with my car and the situation became so bad that there was too much play in the main compressor shaft and hence it destroyed the clutch. I got the compressor repaired and changed the clutch. All this was done in june last year and till now its working fine so I think the work was fine. The new unit will be somewhere around 20k only the compressor and kabli will cost you around 8-10k. Fitting charges and gas etc are extra charges so that will be an additional 5k rs.
But since your problem is just at the mild stage and may be its only the clutch relay that has gone bad change it and if the system works than do get opinions from different AC mechanics. I showed my car to over 15 AC mechanics will the AC was still cooling and they all said chalao jab tak chalta hai phir replace kara laina.