You do have rights as a parent and the child has the right to maintain relationships with his/her father (and there is a visa category especially for this).
You applied for a Tier 2 from within the UK, when you were there as a visitor. You might have been successful had you chosen to apply for an extension to your visitor visa. Making an application in an entirely different category almost invariably invites a negative decision, as you discovered. From that point, your credibility was damaged, and became difficult to repair in its own right. Your subsequent request, the appeal, made matters even worse, as the fraud clause was invoked and you were banned for 10 years.
The family settlement route would not be available as an option: you are married, albeit separated. That the mother is able to travel to your location, as she did for your child’s birth, would not be supportive of a request by you to enter the UK and to lift your current ban.
You have significant obstacles to overcome, best addressed in consultation with those well-versed in immigration law, particularly in complex cases such as yours.
Here are some resources to get you started:
Credit:stackoverflow.com‘
4 Mar, 2024
5 Mar, 2024
5 Mar, 2024
4 Mar, 2024