Robins, who only joined Coventry last September, was appointed Huddersfield Town boss on Thursday after lifting the Sky Blues out of the relegation zone and turning them into play-off contenders.
And chief executive Tim Fisher says the staff and fans are left deflated by the 37-year-old's decision.
Fisher said: "To say we are disappointed is an understatement. We gave Mark a three-year contract which we saw as a long-term appointment.
"Added to this, Mark was able to bring in those members of staff that he requested, plus we gave a considerable financial commitment in terms of players brought in during the loan windows to ensure that we had a strong, competitive team on the pitch.
"Over the last few months, we have had several clubs interested in securing Mark's services and we have fought off multiple approaches. We made every effort to ensure he stayed with us.
"Once it became obvious that there was no way forward we had no choice but to eventually allow him to speak to Huddersfield.
"The club went through a very rigorous process in appointing Mark, we believed in him, his management team and felt he was ideally suited to the challenge that we face at Coventry City.
"The turnaround in fortunes at the club has vindicated both the faith and trust we showed in Mark and I believe everyone, supporters and players alike, agreed we are at the start of something very special and exciting.
"So for this departure to happen at the business end of the season and in this manner, is frustrating and disappointing in equal measure for all concerned. It is fair to say that we are all deflated by Mark's decision to go and, equally, I am sure the supporters will feel the same frustration."