race for (something)
1. To compete (against someone) in a trial of speed or time in order to win some prize or reward. In this usage, a noun or pronoun can be used between "race" and "against" to specify what or whom one is racing. A: "I want that toy!" B: "I'll race you for it!" The two biker gangs are racing for control of this area.
2. To move, run, drive, etc., in a very hurried or frantic manner in order to reach or get to some place. As soon as we pulled into the garage, Tommy jumped out of the car and raced for the bathroom. I raced for the telephone the moment it rang.
race someone for something
to compete against someone for a prize; to try to outrun someone to get to something first. I will race you for the grand prize. Ned raced his sister for the breakfast table every morning.
race for something
to run or drive fast to get to something in a hurry. I raced for the door to see who was there. The children raced for the best seats in front of the television set.