B737 crew returned to gate after engine anti ice valve would not open prior to takeoff. Contract maintenance addressed issue but in process disconnected the incorrect canon plug causing other engine anti-ice anomalies to occur during subsequent takeoff and en-route.

Narrative

Weather at ZZZ was 6C with passing rain showers; ceilings were about 2;000 ft. MSL and vis under the weather was 6-10 miles depending on showers; occasionally about three miles in heavier showers. Local time was late afternoon and takeoff was at night. The aircraft arrived 1+20 late due to operational reasons and we were originally scheduled to depart 1+34 late. Preflight; pushback and engine start were normal. During taxi there was no precip until approaching the runway. Both the FO and I did not turn on the Engine Anti-Ice (EAI) during taxi in error; but as the precip started the FO advised and turned on the EAI. We were then cleared for takeoff; and while lining up the FO stated the #2 EAI light was still bright blue. I advised Tower we had a light in the cockpit and needed to clear the runway. Tower canceled the takeoff clearance and we followed taxi instructions. I cycled the EAI switch with no change in indications and both of us agreed we needed to go back to the gate. I shut down the #2 Engine since there was no EAI available and precip was persistent. Also advised Dispatch of the return; which they coordinated Maintenance at the gate. Contract Maintenance showed about 30 minutes after parking and proceeded with an engine run at the gate with no positive results. Contract Maintainer had a heavy foreign accent and at times was difficult to understand. Maintenance Control advised we would MEL the EAI open. Contract Maintenance completed the process and MEL with Maintenance Control; I checked the logbook and documentation which was correct. Passengers were offered water service and opportunity to deboard approaching XX hours since boarding. Subsequent preflight; engine start and taxi were normal. There was a runway change to XXL and the Departure Plan Checklist was completed. There was no precip at the time but Runway XXL was wet. During takeoff roll passing about 60-70 KIAS the Master Caution light illuminated. I checked the panel and then EAI panel and saw the COWL ANTI ICE light illuminated. We were rapidly approaching 80 KIAS and I advised the FO to continue the takeoff. The remainder of the takeoff and initial climb were normal. Passing 1;000 ft. AGL; as the autothrottles pulled back power about 1.5% to set climb power the COWL ANTI ICE light extinguished. All engine indications throughout the event were normal. After accomplishing all climb activities and passing 10;000 ft. I referenced the QRH for information. The checklist describes using reduced thrust to alleviate an overpressure in the cowl. The COWL ANTI ICE light remained off the remainder of the climb and flight. We did not delete the reduced thrust in the climb. I advised Dispatch of the event and after advising Maintenance Control; they told us Maintenance would meet the aircraft at ZZZ1 and it was ok to continued. Cruise; descent and approach were normal. During landing roll; with reverse thrust engaged; the MC light momentarily flickered but was too brief for me to catch the warning. Taxi in and shutdown were normal. Contract Maintenance proceeded to inspect the EAI and found the Maintenance at ZZZ had disconnected the wrong cannon plug; supplying erroneous information to engine systems. ZZZ1 contract Maintenance fixed the problem and worked with Maintenance Control to document the issue. The subsequent flight from ZZZ1 to ZZZ2 was uneventful. Cause: Faulty contract Maintenance on Engine Anti-Ice valve caused distraction on takeoff roll.Suggestions: Validate contract Maintenance Personnel at ZZZ are competent and properly vetted. This may apply to other stations. Rejecting the takeoff at may have been another choice. However; with a wet runway; at nearly 80 KIAS; continuing the takeoff was most likely the least risk option. The FO and I discussed the scenario. We both agreed we made the correct choice based on the circumstances.