Hello gumbo,welcome here
I suggest as a first step in your preparations you take some time to review the UKVI guidance on Child Settlement here:-
https://www.gov.uk/government/public...children-set07
You'll find some good information of the key aspect of 'Sole Responsibility' too.
There really isn't any written information from UKVI on specifically how to prove sole responsibility status, simply because meaning of the term cannot be precise when each family situation is unique.
Remember though that sole responsibility and sole custody are different.
Sole Responsibility is a UKVI visa requirement.
Custody is a Philippine legal indicator.
You really need to satisfy UKVI that you have been the sponsoring parent for a substantial period of time and that since your migration you have been the main person exercising parental responsibility for the child.
For visa applications this means clearly showing, with examples, that you hold, and have been holding, the ultimate responsibility for all the major decisions relating to the child’s upbringing and providing the child with the care and control and the majority of the financial, health, education, religious and emotional support needed.
Additionally you need to show that child's other biological parent contributes nothing to the child's upbringing in these essential areas.
Consider submitting proof of paying school and medical bills etc
State how you made regular and substantial financial remittances and how you achieved a significant degree of contact and communication.
It's important to understand that financial support alone is not enough. You need to show documented evidence that you've made regular contact with the school, teachers, doctors etc
The outcome of sole responsibility centred visa applications do depend to a large extent on your own unique circumstances, and your ability to provide good supporting evidence.