Opposition parties accused Labour ministers in Cardiff of not standing up for Welsh interests.
Drakeford said: “I cannot say this morning that the answer to the problem is that the Welsh government will provide.”
He also told the Senedd’s Finance Committee that he had some “objections of principle” about diverting money from public services to cover responsibilities that he said lay elsewhere.
Earlier in the session, taking place to scrutinise the Welsh government’s draft budget, Drakeford told Senedd members: “What I am not going to be in a position to do is to provide money to organisations in Wales for which the Welsh government is not being provided with funds.
“Any money we get to cover the cost of public sector employers we will pass directly to those services. We are simply a pass-through for those purposes.
“But I can’t magic money out of the Welsh government budget to cover costs that are not being covered elsewhere.
“I’ve been following what my colleagues have been saying in front of all the committees and this has come up absolutely regularly: ‘How are you going to be able to help this organisation, how are you going to help that organisation?’
“The answer is we are not going to be able to beyond the help we are providing in general.
“In local government there is a 4.3% uplift… I’ve been continuing to have discussions to see if there are any further pressure points that we might be able to assist with, but at that point we will have exhausted the Welsh government’s ability to help.”
