What happened
The FAA’s narrative, in sentence case. The FAA writes it in capitals and removes private details.
During the conduct of a FAR 135.293 check, the PIC was conducting the takeoff portion of a touch and go. The aircraft passed over the arresting gear cable as the aircraft was beginning to rotate for takeoff. The aircraft turned right 5 to 10 degrees and pedal was applied to correct this heading change. The aircraft became airborne immediately. While on the climb out an attempt to raise the landing gear. The decision was made to position the gear in the down position. Immediately on touchdown of the nose wheel we had an indication of a flat tire. The aircraft came to rest on the left side of the runway. Upon deplaning we saw that the nose wheel was turned completely to the right at approximately 55 degrees with what tire that remained wrapped around the strut and the wheel guard was completely twisted around the wheel and strut. It is believed that the aircraft hit a clamp on the cable and this caused the tire to be damaged. The arresting gear had indicated down in the ATCT. Airport maintenance personnel indicated the arresting gear was down.
As the FAA filed it
DURING THE CONDUCT OF A FAR 135.293 CHECK, THE PIC WAS CONDUCTING THE TAKEOFF PORTION OF A TOUCH AND GO. THE AIRCRAFT PASSED OVER THE ARRESTING GEAR CABLE AS THE AIRCRAFT WAS BEGINNING TO ROTATE FOR TAKEOFF. THE AIRCRAFT TURNED RIGHT 5 TO 10 DEGREES AND PEDAL WAS APPLIED TO CORRECT THIS HEADING CHANGE. THE AIRCRAFT BECAME AIRBORNE IMMEDIATELY. WHILE ON THE CLIMB OUT AN ATTEMPT TO RAISE THE LANDING GEAR. THE DECISION WAS MADE TO POSITION THE GEAR IN THE DOWN POSITION. IMMEDIATELY ON TOUCHDOWN OF THE NOSE WHEEL WE HAD AN INDICATION OF A FLAT TIRE. THE AIRCRAFT CAME TO REST ON THE LEFT SIDE OF THE RUNWAY. UPON DEPLANING WE SAW THAT THE NOSE WHEEL WAS TURNED COMPLETELY TO THE RIGHT AT APPROXIMATELY 55 DEGREES WITH WHAT TIRE THAT REMAINED WRAPPED AROUND THE STRUT AND THE WHEEL GUARD WAS COMPLETELY TWISTED AROUND THE WHEEL AND STRUT. IT IS BELIEVED THAT THE AIRCRAFT HIT A CLAMP ON THE CABLE AND THIS CAUSED THE TIRE TO BE DAMAGED. THE ARRESTING GEAR HAD INDICATED DOWN IN THE ATCT. AIRPORT MAINTENANCE PERSONNEL INDICATED THE ARRESTING GEAR WAS DOWN.