Frankfurt distinguishes freedom of action (actions stem from person's desires), and freedom of the will, having second order volitions and enough control to bring first order desires in line with them. He needs this distinction to argue that freedom of the will may exist without freedom of action, which is what a compatibilist needs to reconcile free will with a deterministic world. His interest in the moral responsibility is mostly negative, he wishes to undermine the libertarian principle that one is morally responsible for what one has done only if they could have done otherwise. This is done by presenting thought experiments where a person seems intuitively responsible for their action even if they could not have done otherwise, see Information Philosopher's discussion of the Principle of Alternate Possibilities. The most famous example is Jones who either does on his own what Black wants, or, should he be disinclined, is forced by Black to do it. In the former case Jones seems responsible for his action even though he could not have done otherwise.
Frankfurt distinguishes creatures with second order volitions (persons) from those without (wantons), see Holton's commentary on Frakfurt's theory. Non-human animals and infants are all wantons in his sense. A wanton addict has no wants about his addiction, he is indifferent to being an addict, in contrast a person addict wants not to want the drug, but can not resist the craving. Since animals and infants are typically not held morally responsible for their actions acting on first order desires apparently can not, by itself, bring about moral responsibility. Frankfurt does not directly answer if a person with irresistible impulses is morally responsible for acting on them, but presumably not. It is only if the control of the free will can be effectively exercised that a person becomes morally responsible.
This is not dissociating moral responsibility from free will but rather tying it to free will (in Frankfurt's sense) very closely. What it does dissociate moral from is legal responsibility. For a number of practical reasons we may wish to hold individuals legally responsible regardless of whether they are morally responsible or not. For one thing, we have no reliable practical way of distinguishing between wantons and persons.