Many thanks for your patience. Due to your length of service you will not be in any sort of strong position to ask for more money. That is because if you have been continuously employed at your place of work for less than 2 years then your employment rights will unfortunately be somewhat limited. Most importantly, you will not be protected against unfair dismissal. This means that your employer can dismiss you for more or less any reason, and without following a fair procedure, as long as their decision is not based on a reason which makes a dismissal automatically unfair. These include:
• Discrimination due to a protected characteristic (i.e. because of gender, race, religion, age, disability, sexual orientation, etc.)
• Taking, or trying to take, leave for family reasons including pregnancy, maternity leave, paternity leave, adoption leave, childbirth and parental leave
However, if the dismissal had nothing to do with any of the above exceptions then you would not be able to challenge it. In that case your only protection would be if you were dismissed in breach of contract. That could happen if you were not paid your contractual notice period (unless you were dismissed for gross misconduct) or the employer had not followed a contractually binding dismissal procedure. If you did not have a written contract in place you would be entitled to the minimum statutory notice period of 1 week. Your employer would either have to allow you to work that notice period and pay you as normal, or they will have to pay you in lieu of notice.
So the employer can easily and legally dismiss you anyway, just by paying you the notice period you are due under contract. As long as the settlement covers your contractual notice period and any accrued holidays they are already giving you what you are due under law and you cannot ask for anything on top of that. Whatever they offer you in addition is entirely at their discretion and you have no legal grounds to ask for more.
I'm sorry if this is not necessarily the answer you were hoping for, however I do have a duty to be honest and explain the actual legal position. This does mean delivering bad news from time to time. I hope you understand and would be happy to provide any further clarification if needed. If you are still satisfied with the level of service I have provided regardless of the contents of the answer, I would be grateful if you could please take a second to leave a positive rating by selecting 3, 4 or 5 stars above. Thank you