Can I claim my child who’s mom Is on state insurance? If I do will they lose their insurance?

[Critter-3]

That is a question for the mom to ask the insurance company she has insurance thru.  

[NCperson]

@sal98233 - it might be worth beginning with whether you even control that decision.  If the child lives with Mom for more than 6 months of the year, Mom controls whether she will permit you to claim  the child as a dependent. 

Have you discussed this with Mom and is she ameniable? if yes, then the next question is to the state insurance program.