Converting Battery Operated Motion Sensor Light to AC Powered
A PIR motion sensor works by detecting low-level AC signals coming from the IR sensor, and the design of the light probably depends on the fact that the power is supplied by batteries, which introduce no significant noise into the power.

However, your cheap AC adapter probably has enough ripple and noise that couples into the motion sensor to keep it triggered much of the time. You'll have to reverse-engineer the lamp fixture and add additional decoupling.