How to prove sole responsibility for my children under tier 2 visa

Hello, I would like to seek guidance on the process of bringing my children here in the UK specially in proving my sole responsibility over them. I came here under then the Tier 2 general migrant visa. I am currently on my 4th year of a 5 year visa. I am a single mother of 2, not legally married to my children's biological father, although the father had acknowledged my children and are sometimes in communication with them and visits them at most 2-3x a year also send money sometimes. My children has always been under my care primarily even after we separated (from caring to living expenses). When I came here my children are still under my family's care and majority of my children's care and needs are still under me.
May I know the best possible way to prove that I have the sole responsibility over my children given that they have communication and occasionally receive financial support from him. I am afraid of not being able to bring them over because of this reason specially that my eldest is now 16y/o.
Your help is highly appreciated.

Thank you

See also Sole responsibility.