You say it sounds like a diesel engine at idle? Check the aux belt tensioners
Checked the tensioners, nothing squeaks or is noisy, however recently the car is having a harder time starting like it isn’t getting enough power. I think the alternator might be going, but I am not sure if that is the source of the noise because I would have thought it would be constant throughout driving, not just acceleration.
Could it be a noisy release bearing?
The actual disengagment and engagement of the clutch there is no noise, it is only when I accelerate. So I don’t think so but I could be wrong.
I think I am experiencing a similar problem in my 2010 NC, occasionally from a stand still it sounds like there is some gravel/stone noises coming from the rear of the car, I was planning on changing my diff oil this/next weekend and then the gear oil the following weekend as the vehicle is due a change and I’m not sure if a previous owner has done this.
I was going to change my gearbox oil this weekend and the diff oil the weekend after if it doesn’t improve. Hopefully the problem is just a dried out gearbox being hot and noisy.
To add to that unlike yours my car does not sound like a diesel at idle. So it could be something different or two separate issues.
From what I can find on the forum and other sites this was because I had my AC on in idle, seems to be a little quirk with the mx5.
If the universal joint bearing at the rear of the prop shaft dries out, the needles in there can sound very rough, like crunching gravel.
From what I can find this would be a constant noise, not just on acceleration. I think I will have a look under just incase though.