The firm hopes to run a full timetable next week, but trains may be shorter
First Capital Connect offers a "shoddy" service for commuters using the line from London to Bedford and Brighton, Transport Secretary Lord Adonis says.
Standards are nothing like what "passengers expect", he added, after strikes over pay, driver shortages and disruption caused by the cold weather.
He has demanded "urgent improvements" and did not rule out the option of stripping the firm of its franchise.
The company has apologised to customers "for the recent delays experienced".
"We fully understand the concerns and frustrations of our customers who have experienced disruption for a number of weeks now," said a spokesman.
Lord Adonis described the Thameslink line operated by First Capital Connect as "one of the most important" for those who commute to and from London.
It runs through stations including St Pancras, Farringdon, City Thameslink and Hendon on its way to Bedford and Brighton.
There have been delays and cancellations because members of train drivers' union Aslef had been choosing not to work overtime.
But this week they voted to accept a pay deal, and First Capital Connect said it hoped to run a full timetable again from Monday.
The recent cold snap has added to the problems, with some trains damaged as a result of the weather.
Overall the company has been providing "a very substandard service on the Bedford to Brighton line over recent months," Lord Adonis said.
"They've got to bring about significant improvements on that route," he told BBC London.
"I will look at all options open to me if they fail to do so."
Normally compensation to passengers would comprise tickets equivalent to five return journeys, but the First Capital Connect said a "revised and increased" deal would be offered to commuters "in due course".