Context is queen
What is the context of limiting his time with H? Without that, you do come across as controlling. He comes across as not being truly on board with those decisions and is passively aggressively undercutting you by presenting the decisions as yours alone to H and by lying to you about when he sees her. (Lying by omission is still lying. That is not acceptable.) The limits themselves do not appear unreasonable on the face of it - 2 date nights a week and 3 coffee times are more than many secondaries and couples can manage.
You may well be controlling and he may well be passive aggressive. But I suspect more context will paint a different or at least a more complex picture.
Also how long had he been seeing H? Do you have a friendship with her? Any contact with her at all? Are you seeing others? Sorry for the barrage of questions but I am again seeking more context.
If this is new, then the boundaries are going to be in very much in flux as you and he and she sort things out. Secondaries - aware ones anyway - realize that they are not the first priority - they may not be 2nd or 3rd even. Couples have to set priorities. But there is an important difference between setting priorities and understanding limitations - of time, money, energy, etc. - and dictating to a secondary what the relationship dynamic will be. The first is hard from a secondary's point of view but manageable. The second treats the secondary poorly and denigrates their value as a person and as a person in relationship to one or more individuals in a couple.