I absolutely think an unhealthy relationship affects children. But not just in terms of one boyfriend to another, I also see many married couples in unhealthy relationships where their children are paying the price.
I grew up in one of those homes where my parents thought they were doing the right thing by staying together and 'assumed' we were better off yet I was silently crying myself to sleep each night. They didn't even fight in front us. It was the lack of love and a house full of resentment that affected me.
I think if your friend can find a ground where her children still feel like their mum is happy it might be okay. But I'm not sure how much happiness even she would be feeling yoyoing from one relationship to the other. It doesn't sound like the best environment for anyone involved included her children.