Have you actually applied for a credit card in the UK? You may wish to look at 'credit builder' or 'starter' credit cards, which are designed for people with a limited credit history. See, for example,
https://www.vanquis.co.uk/products/credit-cards/credit-builder-credit-card.
Once you are certain that you are unable to raise the funds from any other source then you might wish to write to GTR along these lines:
Prosecutions Department
Govia Thameslink Railway
Eastside Offices
Kings Cross Railway Station
London
N1C 4AX
DATE
Your ref: REFERENCE NUMBER
Dear Sir/Madam,
I write in response to your letter dated 14 April 2025, which related to several alleged incidents of fare evasion by myself between 14 December 2024 and 24 March 2025.
I am grateful for your offer of a financial settlement to resolve these matters out of court, and am willing to pay the proposed sum of £510.80.
It is, however, beyond my present means to make the proposed settlement in a single payment. My monthly income is £XXX.XX, and I have no capital on which to draw. [ONLY INCLUDE THIS IF TRUE]. I have enclosed recent payslips and bank statements as evidence of the above. No informal financial support from family or friends is available to me. In addition, I have made enquiries as to financing the settlement by credit, but understand that this avenue is unavailable to me because of my short period of residence in the UK.
Might you be amenable to payment of the £510.80 settlement by instalment? By way of indicative terms, I can make an immediate payment of £XXX.XX, and will be able to make a further payment of £XXX.XX in full and final settlement of this matter on DATE.
Your continued cooperation in this matter is much appreciated. I look forward to hearing from you.
Yours faithfully,
XXX
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Edit - I note from your attachment above that you are accused on short-faring on 24 March 2025 - that is, after you were stopped on 21 March 2025. Have you continued to avoid fares after 24 March 2025?