I don't know why so many people here are allergic to specifying which mass they are talking about, which causes all kinds of disagreements over terms and formulae where there should be none.No. Gravity curves spacetime. Light, like mass, follows the geodeisc of this curved space. It is not necessary for light to have mass in order to do this.
Light has no rest mass. It has gravitational mass. In a gravitational field, it has a weight. If you take lights dp/dt in gravitational field, you get a non-zero value. That's a force. That's weight. It's an inertial force, yes, but so is the weight of any other object.