Your income for tax purposes was £163,538 (£162,495 + £593). You are not entitled to a personal allowance for 2013/14 as your income exceeded £118,880. You lose £1 of the £9,440 personal allowance for 2013/14 for every £2 of income over £100,000. That's where the problem is. You were on a tax code of 851L, then it was changed to K92 M1. The M1 part means that any underpayment is ignored until the end of the tax year to avoid a big tax deduction in one month.
The first £32,010 of your income is taxable at 20% (6,402.00), the next £117,990 is taxable at 40% (£47,196.00) and the balance of £12,945 is taxable at 45% (£5,825.25). Your total tax liability is, therefore, £59,423.25. Deduct the tax deducted through PAYE of £56,873.60 and you are left owing £2,549.65.
I have assumed that you did not make pension contributions other than via your salary, if at all, or gift aid donations. You said you appear to be underpaid by about £3,000 but without seeing what other income you have disclosed in the tax return, I cannot reconcile the difference.
You need to decide before the start of the tax year whether to have your personal allowance taken away so as to avoid an underpayment, all else being equal. If your earnings were not what you thought they would be, you may get some of the personal allowance back after the end of the tax year.
I hope this helps but let me know if you have any further questions.