put (one's) weight behind (someone or something)
To use one's influence or power to support or promote someone or something. The small-town candidate is hoping the state senator puts his weight behind her in the race. If we can get a corporation to put its weight behind our charity campaign, we can raise even more money.
put/throw your weight behind something
use all your influence and power to support something: Several of the country’s leading politicians have thrown their weight behind the campaign.