B737-800 Captain reported loss of autopilot and hydraulic system failure caused controllability issues in flight. The flight crew performed a go around; regained aircraft control and landed at destination airport.

Narrative

On left downwind Runway XXC into ZZZ we experienced a loss of flight control effectiveness. Condition were IFR with light to moderate Rain and visibility of 2/1/2 mile at the surface. Pilot flying (PF) was First Officer (FO); and I was the pilot monitoring (PM). We were given a descent clearance to be followed by a base turn to final. While descending on the downwind; FO stated that the aircraft was making an uncommanded turn. With his hands on the control the Autopilot disengaged. FO struggle to regain control indicating strong control force and a pitch down trim force. Concern was a possible runaway trim. I immediately deployed the trim wheel handle and began to trim airplane aft according to FO instruction. My only concern at the time was flying the airplane and regaining control. The only indication I remember up to this point was the red Autopilot annunciator illuminated. ATC called several time to issue turn instruction. We did not respond; due to the urgency of aircraft control. After regaining control; there were concerns about the trim as express by the FO. I decided to eliminate confusion and cutoff the stab trim switches. I informed the FO that switches were cutoff and that we were now manual trim. I then responded to ATC and [requested priority handling]. ATC acknowledged the request and requested the standard passenger count and fuel onboard. I want to land ASAP; now that we had control under what seemed to be manual reversion. The FO was doing a great job turn and descending. We were able to turn and descent to 4;000 ft. and establish ourselves on final. There was much dialog and sense of urgency in our communication in the flight deck. I wanted to land immediately; since time was a factor and weather was not in our favor. As we started to configure with flaps; we attempted to extend the gear. The gear did not extend. At this point it became an approach to landing would not be feasible. I requested a straight out miss and maintaining 4;000 ft. ATC granted that request and shortly thereafter gave us a climb to 8;000 ft. and a turn to the east. At this point; we got busy in a hurry. FO wanted to clean up the flaps. This made sense as first and then I reconsidered. Valuable time was being wasted trying to climb and configure; I also did not see any advantage to clean up with a hydraulic issue. I did not want to lose the use of flaps and then complicate issue more. This was discussed between the FO and I. We agreed not to clean up completely. We left the flaps a 1; the indicator stop a 5. I instructed ATC that we were going to stop the climb at 5;000 ft. They gave us clearance to maintain 5;000 ft. This helped us focus on the gear problem. At this point; I also decided to bring the jump seat pilot into helping us. I instructed him to plug in a headset. Up to this point; everything was just happening too fast and I needed to slow things down and get the gear down. I opened the QRH and started the Manual gear extension checklist. I had the Jump Seat Pilot extend the gear since he had great access to these handles. Right gear was dropped first; this caused a yaw to the right and a pitch down attitude. More manual trim required. Then nose and left gear. This stabilized the yaw. At this point; I was ready to come back for the approach and advised ATC. They started to give us headings back to ILS XXC. I ask the FO is he was OK; or needed a break. He informed me that he was good to continue pilot flying. I then instructed the Jump Seater to monitor FO and keep him focus and talk to him and the same time; I instructed FO to keep the turns to no more than 10 degrees. I advised ATC; that we need a shallow intercept to final. I notice that HYD B system was at 60 percent and brake pressure just above 2;500 PSI. I coordinated with the FO and we agreed that he would only fly the plane. I had the Thrust Levers and the trim. We settled on a speed of 170 kts. for approach. Again; Flap handle was at 1 while indicator was at 5. We broke out at 400 ft. AGL with runway in sight and light showers. Once landing was assured; I set the power to idle. FO needed back trim to arrest the sink rate. I trim back aggressively. We landed and max brake was applied. Control was exchanged and I was able to turn off slowly clearing the runway. We were towed to our gate.Time; Weather and operating in a terminal area under IFR condition with no Autopilot.