In a question and answer session today, Mr Drakeford said he was baffled by the failure of Boris Johnson in refusing to help protect those parts of Wales where there are few cases of coronavirus by banning movement from areas in England with high numbers of cases.
"We will be hammering it home in talks with the Prime Minister and if the situation does not improve, we will take action, we will use our own powers," he said.
"But to have to take action is second best. The simple, sensible way to avoid that is for people from high case areas not to travel to low case areas taking the virus with them."
More Covid-19 coverage:
Mr Drakeford urged those in the shielding category to take extra care during the current upturn in cases but said he was not asking them to shield again. However, he said the Welsh prescription collection service would continue until March and that volunteers were still helping people on the list.
He also answered a question on why the TV Show, I'm a Celebrity was being allowed to continue filming in a lockdown area on the north Wales coast and said it was because the show was providing work, one of the exceptions to movement restrictions in the affected areas.
"It is not the work that you or I am used to but it is work," he said.
"We are in talks with the production company to ensure that the filming is done safely, taking account of the rules."