How can I explain something difficult to my 6 year old?

Hey everyone! This is probably going to be a long post but i want to provide as much background info as I can.

I have a 6 year old bonus daughter. I have been in a serious relationship with her dad since she was 15 months.

We have 50/50 custody with her mom. We all get along good but we have to very different households. At her moms, she is either watching TV or playing on her table 24/7, where at our house we try to limit her screen time and encourage independent free play and outdoor play. We also do alot of crafts, games, and activities together. At her moms house, she eats whatever she wants whenever she wants (she eats alot of junk and candy, even for dinner), where at our house we try and encourage healthy eating choices. At her moms, she can talk like a baby and cry/whine or bable like a baby to get what she wants, where at our house, we encourage her to ask and use her words because she is 6.

Her mom asks us for advice on the time on how to get her to “behave” better. Her mom says she doesnt listen and thay she has to yell at her all day. At our house, we use gentle parenting and related consequences and she seems to respond to that type of parenting very well. She obviously still has her moments (what child doesnt). The only issue we have with her is she will not play independently. I am a stay at home so I do crafts all day with her, she cooks with me, we play games together, go for walks, go to the park, but the second I tell her it’s time to go play, she refuses. She will just sit on the floor or lay down or follow me around.

Three weeks ago, I started babysitting a 3 year old girl. They play really well together. Ever since I’ve started watching that girl, when she goes home, my bonus kiddo cries, usually until bedtime. She won’t play, she won’t do anything. I try and talk to her about what’s wrong and what I can do to help but all she says is " I miss mom". She wants to call her mom and cries because she misses her mom anytime she is asked to play independently or is given something for dinner she doesn’t want (I can’t even get her to eat chicken nuggets). She never talks about missing her mom when the babysitting kiddo is here or if I’m cooking with her, or we are at the park, or we are swimming at the lake. She only cries about missing her mom when she’s asked to do something she doesn’t wanna do.

My question is: how do I explain to her that it’s okay to be sad and miss mom, but she can’t use it as an excuse when she doesn’t wanna do something? I feel horrible if I’m giving her the impression that she’s “in trouble” for missing her mom, but I also feel like she uses “missing mom” as a way to get out of doing stuff. Any help or advice would be appreciated as her dad and I don’t know what to do.