have (one) on the run
To have one in a vulnerable position. After scoring five goals in the first period, we really had our opponents on the run.
have someone on the run
If you have an opponent on the run, you are performing better than them or are in a stronger position, and can control their actions and defeat them. It is clear that the Opposition thinks it has him on the run. We've got the Government on the run and we'll keep them on the run. Note: You can also say that someone is on the run. They sensed their opponents were on the run.