Hi,
Please re-read my last reply (Post #3).
I have unplugged the transmitter from the rest of my circuit and have connected / powered it up separately and tested it on its own and the buzzing is still there. So the transmitter is interfering with itself - somehow. So I have established it has nothing to do with the rest of my circuit, including the audio Amp, which is all on a separate main PCB board and has been ruled out through process of elimination.
I have posted the schematic and layout for the transmitter module break out board (which I designed and built) but I cannot post a comprehensive schematic of the actual transmitter module itself because I can't find one. All I had to go on was the information I could piece together from what other people had found out through trial and error.
The transmitter is made by a company called Dorji and the Model is DRA818V. It is a pre-built, self contained VHF FM Transmitter module based on the RDA1846 FM Transmitter IC and is more or less a cut down version of a Baofeng Mobile Radio in a small module package.
This is a typical connecting diagram for the DRA818:-
The only real difference between the breakout board I made and that typical connecting diagram is that my breakout board doesn't have a LPF (Low Pass Filter) in series with the RF OUT (Pin 12).
And the reason I never included a LPF in my design was because when I was prototyping on breadboard in the early days of my design, I never had or certainly never noticed this buzzing so made the assumption that everything was good. But since I built the little break out boards for the transmitter module, the problem (buzz) has become extremely bad. Most likely due to the proximity of the antenna to the rest of the circuit without adequate filtering.
Researching in the Early Days:-
When I was researching these transmitter modules in the early days, I found lots of designs for Arduino APRS trackers using these modules and the majority were not using any kind of filtering at the RF Output so that combined with not having any issues while prototyping on breadboard seemed to validate my lack of need for any filtering - until now.
Since having my problem with this transmitter buzz, I've found other break out boards which have used vias stitched all round the perimeter of the RF PCB Trace to form an electrical RF screen, which is another thing I didn't do. I had thought that keeping the trace / wire on my breakout board short and the connection as close as possible to the RF OUT pin on the transmitter would prevent any RF leakage.
To Answer Some Questions:-
Yes, the antennas I am using are tuned. They are commercially built, dedicated Amateur Radio Antennas. The main antenna I have been testing with is a "rubber duck" style walkie talkie antenna for 2m/70cm (VHF/UHF) but I have also tested on a dedicated, externally mounted, mono band vertical VHF antenna with the same results.
The frequency range I have the transmitter working at is 144Mhz to 146Mhz (2M Amateur Band) - and before anyone points it out, yes I do hold a valid and current Amateur Radio License in my country so I'm not wildly and randomly transmitting where I shouldn't be 
lol
What the Project is About:-
For those of you that are wondering / interested in what I'm doing / trying to achieve - well, primarily it's a project of learning and boy have I learned along the way!
Each new project I think up, I like to be more challenging than the last so I am forced to learn new stuff while putting it into practice. In at the deep end, sink or swim as it were. Usually I try and build on projects using components and modules I've gained experience from previously and then ramp it up a notch.
In the case of these transmitter modules, I've gone from previously creating a simple Fox Hunt Transmitter which worked without issue, to trying to build a small, portable VHF Repeater which is being quite problematic due to its complexity.
But on the flip side of all the issues I've had getting this far, I've learned so much! I know more about RC, LC, LR Filters than I did before I started and I've never so much as seen an Op Amp before I started this so I know far more about them than I did previously.
I am hoping to go for my full license next year but I'm not a fan of just read and repeat learning. I like to get hands on, see what things do, ask why and find out why by putting it in to practice. I find I learn far more when I'm building stuff and troubleshooting it than I do from just reading the course materials.
And now I'm so tantalizingly close to finishing this part of my project if I can just cure that annoying buzz.....
My next step is to de-solder the transmitter module from the breakout board and fit it on to a new breakout board which has a LPF in series with the RF Out and see if that helps.