The Super League club confirmed in a statement that the 23-year-old failed a drugs test following the 38-6 win at Huddersfield Giants on Thursday, March 16.
He has not been suspended but given an “extended period of leave” while awaiting the outcome of a UK Anti-Doping investigation.
In a joint-statement, former Leeds Rhinos player Minns said: “I want to personally apologise to everyone involved who I have let down and disappointed.
“I admit to taking a banned substance on Mother’s Day 2018, which was a hugely difficult day for me following the passing of my mother last year.
“I was low and depressed on the day and I felt at the time I needed to do it to get through.
“I now totally accept that it was a huge mistake which could cost me my career. I can now only ask for forgiveness and for my arguments of mitigation to be heard and understood.
“I do not wish for sympathy. All I ask is that I am listened to and I can make everyone aware that I did not take this substance to enhance my performance at rugby in any way.”
Minns has been an excellent signing for the East Yorkshire club since arriving from Leeds at the end of 2015 but becomes the latest Super League player to fail a drugs test.
Rangi Chase was handed a two-year ban in November for cocaine use and fellow Man of Steel and England international Zak Hardaker is still waiting to hear what his suspension will be for the same offence after the Castleford Tigers full-back’s positive test came to light just two days before the 2017 Grand Final.
Tenth-placed Hull KR are struggling in their first year back in Super League after relegation in the 2016 Million Pound Game.
Minns was a late withdrawal from Saturday’s loss at Warrington with no reason given at the time.
Rovers chief executive Mike Smith said: “Whilst we are obviously disappointed by this (failed drugs test) as a club, we feel it is very important to stress that this is not just a difficult time for Hull Kingston Rovers, but also for Thomas himself.
“Thomas has expressed his regret and spoken of how he has found life away from the sport of rugby challenging in recent times, particularly since the tragic passing of his mother last year.
“Thomas has been a much valued member of the Robins squad and is a well-liked young man by all at the club. These are obviously difficult times.”
Smith continued: “Following discussions with him since failing the test, we are clear that our priority and focus at this time must be to offer him our support.
“As yet Thomas has not been suspended by the club, but has been given an extended period of leave. He will plead guilty to the offence and ask for a hearing to mitigate his penalty.
“We will be speaking to the relevant bodies and the RFL about how we can now best support Thomas at this time.
“Any decisions over his playing future will be considered at a later stage and are not a matter for now.”
“As a club, Hull Kingston Rovers prides itself of the unity of its playing staff and our commitment to player welfare.
“That is why our priority at present is to ensure Thomas is provided with the support he needs.”
Rovers visit Oldham in the Ladbrokes Challenge Cup fifth round on Sunday having also seen captain Shaun Lunt ruled out for two months with a knee injury suffered at Warrington.