//For example: Total Income £20,500. Less Tax-Free amount of £13,750 =£6,750. On this amount I would pay tax at 20% i.e £1,350. I know where the £4,587 comes from but not were it fits in.//
Where it comes in is that your tax free allowance is £4,587. You can forget about the £13,750 altogether. That is the "starting point" if you like, but the end point is £13,750 minus your state pension (which I assume is £9,163). When calculating your tax bill you don't consider your State Pension as income (becuase it has been paid tax free but your tax allowance has been reduced to cover the tax). So with a total income of £20,500 (including a State Pension of £9,163) the amount considered for tax is (£20,500 - £9,183), or £11,317. Your personal allowance is £4,587, meaning you must pay tax on £6,730. At 20% this is £1,346, which is not unadjacent to your figure.