Do NOT take a lower sum! Why should you? As it was she had an interest free repayment schedule of just over 12 months! You will easily win this in a small claim.
Where you plead a contract, you need to say if it was reached orally or in writing. If orally, you need to state where, when and how it was agreed - this is what the court rule says:
"7.4 Where a claim is based upon an oral agreement, the particulars of claim should set out the contractual words used and state by whom, to whom, when and where they were spoken.
7.5 Where a claim is based upon an agreement by conduct, the particulars of claim must specify the conduct relied on and state by whom, when and where the acts constituting the conduct were done."
So you must say for instance roughly what date the agreement was reached (you say "on or about x date"), where you were and what was said (and refer to any evidence there may be). Go on to say what instalments were paid and when, and what she is in breach of (ie failed to make the instalments due on x, x and x, and has now refused to pay any further instalments).
If you want to read the rule and accompanying practice direction which supplements the rule, here are the links:
www.justice.gov.uk/courts/procedure-rules/civil/rules/part16
and
www.justice.gov.uk/courts/procedure-rules/civil/rules/part16/pd_part16
The language used is a bit "stiff" so don't panic, it's not as tricky as it may appear at first. Read 16.1-4 of Rule 16 and paras 1-3 and 7.3-7.5 of Practice Direction 16.
Something slightly confusing is that the rule provides for you to either include the particulars of claim with the claim form, or within 14 days later. With a small, simple claim like this you'd include the particulars in the relevant box, with no separate particulars to follow (that provision is for more complex claims where the particulars may run to many pages). So ignore all the "if the particulars of claim are served separately...." bits in the rules.
You should say if you are seeking interest (you are entitled to do so). Calculate that at 8% on a daily rate. So if she owes £420 then the interest is 9.2p per day, so you calculate when she should have paid you and the number of days since x9.2p and the continuing at the rate of 9.2p per day. Just leave it out if you don't want to claim it (it's probably peanuts).
The wording to use at the end of the claim where you have to confirm what remedy you are seeking is that you are seeking damages for breach of contract in the sum of £420 and costs (you can only claim the amount of your court fee and I think you can claim loss of earnings for attending court and travel/parking costs.
The rule/practice direction specific to small claims is here -
www.justice.gov.uk/courts/procedure-rules/civil/rules/part16/pd_part16
and
<a class="break-all" href="https://www.www.justice.gov.uk/courts/procedure-rules/civil/rules/part16/pd_part16justice.gov.uk/courts/procedure-rules/civil/rules/part27/pd_part27" rel="nofollow" target="_blank">www.www.justice.gov.uk/courts/procedure-rules/civil/rules/part16/pd_part16justice.gov.uk/courts/procedure-rules/civil/rules/part27/pd_part27
If you google it, there will definitely be government guidance to MCOL and how to make small claims and fill out the forms.