Yes, PUSHED is a Scrabble word!

Letter values in Scrabble US: P 3, U 1, S 1, H 4, E 1, D 2.

Is it a valid word?

PUSHED is a valid word in the official Scrabble dictionary (US), the UK Scrabble Dictionary and Words with Friends (ENABLE).

move with force, "He pushed the table into a corner"
press, drive, or impel (someone) to action or completion of an action
make publicity for; try to sell (a product)
strive and make an effort to reach a goal
press against forcefully without moving
approach a certain age or speed
exert oneself continuously, vigorously, or obtrusively to gain an end or engage in a crusade for a certain cause or person; be an advocate for
sell or promote the sale of (illegal goods such as drugs)
move strenuously and with effort
make strenuous pushing movements during birth to expel the baby

Also: bear on.