Head to the swing bridge by Tobias' Shop to start the mission and then hack the bridge before crossing it. There are three spots you need to investigate to restore power.
Head straight from the bridge until you reach two containers, one blue and one red, and then grab the Audio File: Engineers 01 on the other side. Backtrack east to the first part of the buildings and then head north between some containers and a fence to reach a scissor lift that will take you to the roof. The gate is locked but you can easily head around to the right of the fencing and jump from camera to camera to unlock the gate. From here you should have no issue reaching the first generator.
Head south past all the containers and when the waypoint is directly east of you, down an alley, take a left and get the Audio File: Engineers 02 in hiding at a dead end. Head back a bit and look up to find a stacking crane that you can hack to hitch a ride up. Exit the other side of the container and turn left to climb over another container and drop on the other side. Here, another stack of containers will provide you a way up to the roof where you can hack another camera to open the gate and then ride down to the next generator.
Head south to the locked gate and hack the cameras as you follow the data flow to the unlock panel. Once the gate is open, head inside and turn right and climb up on top of the white container before hacking the crane. Hop off the crane and stop to the left of the open green container and look down to get the Audio File: Engineer 03 . Now head through the green container and turn on the final generator.
Note: There are 3 more Audio Logs available on Abbot Island that you can get now or come back for later. See Engineer Logs for more information.
Head back outside the gate to the container and enter to gain access to the bunker. Inside, hack the laptop and then turn on the power for Clara to complete the mission.