Adomah has signed a three-year deal at the Riverside and Boro are expected to announce his arrival on Wednesday, the day after their embarrassing League Cup exit to Accrington Stanley.
Mowbray has lodged a bid for Leeds United's Ross McCormack and admits his side are short of goals.
"Albert is not going to head balls in the box, or head balls away from the net," said the Middlesbrough boss.
"He is a creative wide player who can get up and down the right wing, he will cause problems to teams with his speed and trickery.
"He can create chances but ultimately we need to have somebody who can stick the ball in the net. We are looking hard to try to adjust that situation."
Leeds boss Brian McDermott has revealed that he has already turned down bids in recent days for some of his players.
"We've had a couple of bids for players which we've rejected," McDermott said.