Let's ignore the £3700 to start with.
You've described claiming that you've got a university degree as a 'white lie'. I disagree (and I'm sure that a magistrate or judge would too).
Saying that you'd got a grade B at GCSE Maths, when you only got grade C, might be regarded as a 'white lie' (although, if it's done to make a gain, it's still a criminal offence).
However falsely indicating that you've undertaken several thousand hours of advanced study, and then passed rigorous examinations, in order to obtain a degree is (when done for the purpose of gain) far more than a 'white lie'; it's a serious criminal offence:
https://www.legislation.gov.uk/ukpga/2006/35/section/2
So there's absolutely no doubt whatsoever that you're GUILTY of 'fraud by false representation'. Any other plea would be madness (and would only increase your sentence).
I'll continue ignoring the £3700 for the moment and address the issue of the £5000+ which you think that you're owed. If you were to take your claim to the County Court the company you worked for might well have a good argument for showing that they shouldn't pay part, or any, of that money to you. Your contract with them was based upon their assumption that they were getting the services of a university graduate but (because you've not got a degree) you failed to provide such a service. So the firm could argue that your contract was invalidated and that they thus don't need to pay you. (It would need someone with more knowledge of civil law than me to accurately assess how a court would respond to such an argument but it's possible that the ruling might be that the firm doesn't owe you a single penny).
Now let's get to that £3700. Where a person is charged under the Theft Act 1968 (rather than the Fraud Act 2006) there's a statutory defence to the charge if the accused person can show that he "appropriated the property in the belief that he had in law the right to deprive the other of it". So, if you were charged under the Theft Act, saying that you honestly believed that you had the right to hold onto the money because it was just part of what was owed to you should be a sufficient defence. There's no similar provision in the Fraud Act but, if your case was well presented to a court by your solicitorr, the court might be minded to accept your explanation for hanging onto the money when passing sentence.
'Fraud by false representation' is an 'either way' offence, meaning that it can be dealt with entirely within a magistrates court or (in more serious cases, or when the defendant demands trial by jury) passed up to the Crown court. It appears unlikely to me that your case would reach the Crown court.
The magistrates' sentencing guidelines can be viewed here:
https://www.sentencingcouncil.org.uk/wp-content/uploads/MCSG-April-2017-FINAL-2.pdf
(See p.281 et seq)
'Culpability' is likely to be viewed as on the borderline between categories 'B' and 'C', with 'Harm' being assessed as category 4. That means that a custodial sentence (for a 'B4' offence) is a theoretical possibility but, with an early guilty plea from a first-time offender, actually
extremely unlikely. A Community Order (perhaps involving a period of unpaid work) would seem to be a far more likely outcome.
A note of caution for the future: Assuming that you're given a Community Order, your offence will become 'spent' one year after the end of that order. From that point on you won't have to declare it on job application forms (or similar documents) unless you're applying for 'exempted' work (such as working with children or vulnerable adults). Up until that time though, if you're asked about whether you've got any criminal convictions you MUST declare your conviction. If you fail to do so you'll again be committing 'fraud by false representation' and, if you're found out, you won't be given a second chance. You'll almost certainly be sent to prison.