In relation to a new partner becoming part of an established lone parent family. "Oh he/she are really good, they took my kids on".
Just makes me think of onerous and unpleasant jobs or tasks that no one wants to do. "Oh alright I will take it on".
If you insist on using it then equally your children have "taken them on" too surely?
No one takes my dc on, anyone I decide to be with would be lucky to have them in their lives.