Cessna 172 pilot reported loss of directional control during landing roll and initiated a go-around during which pilot heard a bump sound. After return to base pilot learned they had struck a runway light resulting in aircraft damage.

Narrative

Chain of Events:I am a newly rated Private pilot as of Day 0. I was doing a solo cross country from ZZZ1 to ZZZ in a Cessna 172 M that is owned by my flight school located in City X. I was using flight following on my way over. I obtained the weather and the winds were favoring runway XX; if I remember correctly winds were from 260 @ 4 knots. I communicated on the CTAF my intentions. I did one touch and go on runway. I maintained centerline and control of the aircraft. I wanted to do a second touch and go to make a softer landing. I came into land the second time for runway XX; I touched down on centerline and completely landed the plane. I felt much happier with that landing. I failed to use right rudder to directionally control the airplane after touch down to maintain centerline and started veering to the left of runway XX. I became frightened and my first instinct was to go-around. I put in full power with 20 degrees of flaps without right rudder correction and unfortunately created more left turning tendency; which in turn created a sharp left turn. I took off the runway on the left side of runway XX; I had just lifted off the ground when a few seconds later I heard a bump near the aft underside of the aircraft. I incorrectly assumed the noise was from the tires spinning still from take off. I departed ZZZ toward ZZZ2 for fuel and then ultimately back to my home airport of ZZZ1. The flight performance and controls felt normal per usual.I did not report the incursion originally because I did not believe there was an issue with the aircraft. The next day I received a text message from an employee who sent me a photo of the right underside of the elevator with a small puncture and yellow paint on the normally white surface; asking if I knew anything of the damage. The damage was discovered from a CFI doing a preflight on the plane the morning after my flight. I realized that something more serious occurred from that bump than my poor assumption that the wheels were just turning. Especially since I did not identify damage in my preflight the previous day from the elevator departing ZZZ1. After the damage was discovered from the instructor the plane was immediately grounded. Fortunately; after I returned the plane no flight occurred with the damaged elevator.After seeing that text message I immediately called my flight instructor and told her the situation; she informed me it would be best to talk to one of our assistant chief pilots. I did so immediately and explained the situation. I filled out an insurance form claim for the airplane. Our chief pilot was informed. After discussion with maintenance and the assistant chief pilot; since I was unsure of the exact cause of the damage; it was speculated that FOD played a factor.On day 1; I received a call from our Chief Pilot; Assistant Chief Pilot; and another person of importance; who informed me that I came in contact with and damaged a runway light at ZZZ and that the City Y will be sending me a bill to replace the light. In order to fly solo again with my flight school at ZZZ1; I was to go up for one flight with cross winds and do touch and go's with a CFI who is not my primary CFI. That was completed today on Day 3 and I was found to be proficient in total winds of 26 gusting to 29 with a crosswind factor of 11-13 knots. HUMAN PERFORMANCE CONSIDERATIONS:I was extremely shaken up by this event and after discussing it with several CFI's we found a variety of factors that played a role in this situation.1. Failure to maintain directional control upon roll out and counteract the left turning tendency of the plane.2. The incorrect to decision to attempt a Go-around when the plane had fully landed with 20 degrees of flaps.3. Complacency - This is not continuing to keep a checkride standard a few weeks after my PPL checkride4. Psychological Defense Mechanism-In this instance I knew deep down something had occurred but I kept trying to convince myself that the bump I heard was from my wheels turning from take off. I kept trying to push the reality and gravity of the situation out of my head and convinced myself it was okay. It was not okay. I should have not flown the plane. I should have reported immediately if I heard an unknown noise once I was at my flight school so the plane could be grounded and checked out. Even if there was not any external damage to the plane I could have damaged something internally. Not only that someone else could have flown the plane after me unknowingly the plane was damaged and could have cause serious injury. It was selfish act in order to protect my own ego.5. Get there it.I was scared and wanted to be back home at ZZZ1.6. Invulnerability hazardous attitude I learned on the ground the different types of accidents that occurred and I tried to be as humble as possible. If I am honest with myself though I thought I would be able to react properly in a situation such as this; I thought that I had a higher standard because I just came from a checkride despite being a low time pilot. I realized and accepted that I have the invulnerability hazardous attitude.7. Previous experiences with lack of right rudder usage.- There had been two previous experiences in my formal PPL training with two different instructors that that lacked enough right rudder usage on touch down; fortunately in those instances they took controls and there was a wide enough runway to correct. I thought I had moved on from these instances but obviously I did not.Overall; what happened on this takeoff was minimal in consideration of what could have occurred. I am extremely grateful to be safe and unharmed and walk away; with a small puncture in the elevator and a damaged runway light. I fully understand and realize that this could have been a terrible accident and a possibly fatal mistake. I realize the extreme gravity of the situation and have a deep remorse of the event that occurred in City Y. I knew that general aviation can be unforgiving. However; on that day I was able to internalize how extremely unforgiving general aviation is.WHAT I AM DOING TO PREVENT RECURRENCE:1. I am currently rewriting my personal minimums form for short and narrow runways.2. I did my primary training in a 160 HP 172 N model. This is a 180 HP M model C172. Not to make an excuse; but I was not mentally prepared for touch and go's in this model because it does act slightly different. Before this flight I only had 2 hours in the M model.3. I will be continuing to train with a CFI who is not my primary instrument CFI every other week to two weeks on touch and go's and ensuring I am not becoming complacent in my rudder usage. 4. I am rereading aerodynamics in the PHAK and will be doing more in depth research on touch and go's and 'the point of no return' for a touch and go. 5. The next solo flight I take I will be talking out my intentions with an instructor and do warm up touch and go's at my primary airport to prime my brain. 6. God forbid this were to happen again but if there is any unknown noise; and I question it for a second I will be landing the plane as immediately and safely as possible to do a high quality check; if there is any doubt I will ground myself and the plane immediately.