“If you do X, I will give you Y”

Offering a child a small treat (whatever it may be) might seem unobjectionable, because parent/carer and child both ‘win’. But what’s really happening here is that you’re implicitly teaching them that if they want something that seems appealing, to get it they will have to do things they don’t want to do. You’re essentially supporting a pattern of exchanging one’s happiness/dignity/ethical standards/morals for some prize.

How can we express approval when our children do something good without manipulating them by implying that we would disapprove if they had made a different choice?

The kind of expressions of approval that are not manipulative are the ones that bubble out of you without any forethought. Anytime you are wondering if what you were planning to say might be coercive approval, it probably is. Is what you are saying the kind of thing you would naturally say to an equal, a friend, or your boss, say? Or does the idea of saying this to your boss seem highly inappropriate?