I recently left the RAF after a long "career" as aircrew and before I left I used my resettlement money and annual educashun money to qualify as a bookkeeper. I am now self employed and one of the things I do is personal tax. Thats my credentials out of the way.
Yes indeed you can claim back the excess. The way it works is that if you travel on duty and get a mileage allowance from your employer for using your own car, that is below the chancellor's allowable 40p per mile you can claim back tax on the difference. Duty does not include home to duty (or whatever it is called in the 21st century), or money paid under get you home schemes. You can claim for up to 10,000 miles per year.
For instance if you do 1000 miles and get paid 25p per mile, you will get £250. The chancellor would allow you to have £400 (40p per mile). You therefore put the difference (£150) in box 2.5 on your short tax return and you will get £150 extra tax free for that year. Note that you dont get the £150 back, but the tax on £150, so how much you get will depend on whether your top rate of tax is 22p or 40p in the pound. It may not sound much but anything that saves you money is a good thing.
I dont think you would need SAMA printouts (although if you have them all well and good), but you will need to keep good records. I would advise a copy of the claim and the related payment slip as the minimum. For JPA, a printed copy of the claim and the payment advise will suffice. Forgive me if I have got the JPA thing wrong, but I left just before it came in.
Hope this is useful.
Shadwell