Firstly, if you had a private arrangement with the mother for the maintenance, and you have stopped it, she will very likely make an application to the Child Maintenance Service (CMS) for a formal agreement to be set up.
It is important to note that child maintenance and whether or not you see the child are not related and are very separate when it comes to the mechanisms of dealing with them. Therefore, if you don't pay the maintenance to the mother, CMS will order you to make payments and will re-assess the amount you are paying too.
Secondly, spending time with your child is very important and the mother should not stop you from seeing her because you feel it's best to reduce the time. I think the reasons you have given are reasonable - and a court would likely agree that once a week and then every other weekend is probably a good balance of time for you to spend with your daughter.
It may be better for you to get a formal child arrangements order in place in order to stop the mother from thinking she can stop contact as and when she likes, but before you can go to court you are obliged to try a mediation information and assessment meeting first.
You can find a local mediator here: https://www.familymediationcouncil.org.uk/
If mediation does not work, you can apply to the court (the mediator will sign your form in the relevant section to prove you have tried mediation first). You need a form C100, which you can find here: https://assets.publishing.service.gov.uk/government/uploads/system/uploads/attachment_data/file/734888/C100_eng_0818.pdf
The fee to apply to court is £215, but if you are on a low income or benefits you may be entitled to fee remission, see here:
You need to file the application in the family court closest to where the child lives.
Does that help?