I agree there is certainly a possibility that a future employer might ask about reason for leaving and might ask about whether there were any live disciplinary proceedings. Obviously it's fairly common for people under an investigation, especially a very serious one, to resign before they are dismissed.
If a future employer asked that question it would be fine for your employer to give an honest answer.
If you genuinely can't even think what the problem may be, then I'd be inclined to stay and clear your name tbh.
if this was my place of employment and I handed in my notice whilst under investigation, then this would be recorded in the question on any subsequent references - "were there any disciplinary action?" - it would remain an open disciplinary. I suspect future employers would see this as guilty as I'd left without completing the disciplinary procedure.
Of course you can leave at any point, just consider the impact on future employment if that is going to be necessary.
do you have a direct referral system to an occupational health nurse? maybe self refer and talk everything through.
being informed verbally that you are under investigation is not sufficient - you need it in writing. I would find your disciplinary policy and ascertain the correct procedure. then work out if this is being followed correctly.
I really need some advice... I have been informed verbally that I am under investigation at work for something to do with misusing the flexi policy, I think it may be classed as fraud but I am still awaiting a letter.
I have been so unhappy at work I have actually been considering handing my notice in before I was aware of this, I just cant take any more stress and I cant function after today. I will face any questioning and take any punishment but I want to leave, can I do that?
I cant think of what it is they are accusing me if and if I have done anything wrong I certainly havent done it deliberately, I have never been through anything like this and am devastated.