Hermione - I think littlebit means that, if you are asked to sign it there and then without legal advice, you could do what she suggests.
Personally I wouldn't. If you go to tribunal, your compromise agreement money will generally be deducted from any award. But you will have a very pissed off ex-employer who will likely make the litigation very difficult for you, very stressful and (if using a lawyer yourself who isn't covered by insurance or no-win no-fee) very expensive. You're better off just negotiating a deal you are happy with at the outset. Also, although not effective for statutory claims, you can arguably waive certain contractual rights in the type of agreement you refer to OP (things like bonus entitlement or notice pay potentially and dependant on the circumstances). So you could sign away something valuable whilst thinking you were 'safe'. I just wouldn't go that road if I were you.
Oh, and another thing, if you negotiate a CA you can add in lots of useful things, like references and outplacement that probably aren't in the first draft and don't get awarded by tribunals, so you might lose out on 'stuff' anyway.
If put under pressure to sign, I would just be very clear that you aren't happy doing it on the spot, need time to think and that you thought you needed to see a lawyer for the whole agreement to work (assuming you think that they are fully aware of the last bit and just trying it on).
On the tax fedup is right, but the rules can get complicated, particularly as to whether payment for notice periods you don't work is taxed or not.