>>>as April is the first month of the new tax year, HMRC assume we are going to be earning this higher salary every month
Effectively correct.
Your personal tax allowance is £10600 per year, which equates to about £883 per month. Let's say, for example, that you'd normally get paid £2000 per month. You'd then be taxed (at 20%) on £1117 (= £2000 - £883), meaning that you'd pay about £223 in tax (which is around 11% of your gross pay).
However if you received a healthy £1000 bonus this month, you'd be taxed on £2117, meaning that you'd pay about £423 in tax (which is around 14% of your gross pay).
i.e. every pound of your bonus is taxed at 20% (because you've already used up your tax-free allowance), unlike every pound of your regular pay (which gets that tax-free allowance).
At the end of next month the PAYE system looks at how much you've earned so far in the tax year, deducts 2 x £883 (as your tax-free allowance) and then calculates 20% of what's left. That's the TOTAL tax you'll owe at that point, so the amount you paid this month is then deducted in order to work out what you should pay next month.
The same thing happens the following month (with 3 x £883 being used as your tax-free allowance so far in the year), and so on. By doing it that way your tax will AUTOMATICALLY adjust itself so that, at the end of the tax year, you'll have paid tax at 20% (or higher, if appropriate) on everything above £10600 but no more.