The former Sporting CP wing-back has been plagued by injury since joining in the summer.
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Rosenior revealed on Friday that Essugo is again sidelined after a new setback: "Dario has run into a very unfortunate injury.
"He slipped for training and he's going to sit out for over a month. It's a shame.
"I've really looked forward to working with him because he seems like an amazing player."
Essugo is yet to make his senior debut for Chelsea.