I’m contracting at the moment, moving from one contract to another with the same company but PAYE with their outsourced recruitment firm.
Somehow, and I’m not sure why, because I’m usually on top of these things, I’ve been on emergency tax for the last 4 months, since I started with the recruitment firm.
I’ve dug out my P45 to get this sorted, but what’s the chance of me having been overpaid, rather than underpaid?
If I’ve been underpaid then I guess an adjustment will be made and I’ll be paid a little more. If I’ve been overpaid then that’s not so great.
What’s the chances of having been underpaid rather than overpaid?

Depends what you are paid. You don’t get any tax free allowance, but you are taxed at a flat rate.

Not an accountant but missus was on an Emergncy code for over two years!

You’ve probably been paying tax on everything you earn. You’ll be owed 20% of the tax free amount (just over a grand). You’ll either get this in the pay packet after they sort your tax code out, or as a cheque in the autumn after the end of the tax year.

What tax code is shown on your P45? And on your payslip?

And was there a gap between employments?

Aye – emergency tax usually means no tax-free allowance. If you’re a basic rate taxpayer, you will have over-paid and either get a refund or, more likely, a coding change to reflect this and balance it up. However, if you are a higher rate taxpayer, you may have under-paid.

By emergency tax code do you mean BR, D0 or 747L W1?

BR means you’re taxed on all your earnings at 20%
D0 means you’re taxed on all your earnings at 40%

Either of the above means that you’re loosing out on your annual tax allowance, in which case contact HMRC to find out why as your previous employer may not have submitted your P45 so HMRC thinks you two jobs. You can ask your employer to submit a P46 for you which will put you on a week one basis, see below

747L W1 means that you’re taxed as if as if each week is the start of the tax year. This is normally because HMRC haven’t informed your employer of your new tax code and you didn’t supply a P45.
Your tax normally works out about right for the year but still worth checking.