Not sure about any audible or tactile "click" on shifter arm...
This mower is right at 15 years old, and service parts are no longer available, sorry. Honda does offer a whole new transmission, but list price is $336 (ouch).
Here's the procedure to remove the trans:
1. Shift the transmission into 2nd gear. Loosen the clutch cable lock nuts and remove the clutch cable from the handlebar.
Loosen the shift cable lock nuts and remove the shift cable from the handlebar bracket. Leave the shift cable connected to the shift lever.
2. Drain the fuel from the fuel tank and carburetor. Drain the engine oil and turn the mower upside down. Rest the handlebar on a support to prevent damage to the recoil starter cover.
3. Remove the rear wheels, drive gears, ratchets keys, and ratchet return springs. Remove the wheel covers and the rear deck scroll (this is the removable section in the rear of the deck...three bolts, as I recall).
4. Insert a screwdriver or rod into the cable holder bracket hole and pivot the transmission to remove the belt from the pulley.
5. Remove the 13 mm circlips from the axle.
6. Remove the 22 mm circlips from the inside of the rear adjusters and remove the adjusters.
7. Loosen the shift cable lock nut and remove the cable from the transmission bracket. Disconnect the cable from the handlebar.
8. Remove the transmission from the mower.
9. Disconnect the shift cable from the transmission.
1. Install the clutch cable to the transmission. Connect the shift cable to the transmission change arm as shown. Do not position the cable on the cable bracket. Install the rear tensioner spring plate in the mower deck and connect the tensioner spring to the plate.
2. Position the 22 mm circlips and washers on the transmission axle and install the transmission in the mower. Route the clutch cable through the hole in the mower deck. Route the shift cable behind the engine.
3. Apply general purpose grease to the inside of the rear adjuster bushings and install the rear adjusters. Install the inside washers and circlips on the adjusters.
4. Install the 13 mm washers and 13 mm circlips.
5. Connect the tensioner spring to the transmission.
6. Insert a screwdriver or rod into the hole, pivot the transmission, and install the belt over the pulley.
7.Connect the clutch and shift cables to the handlebar. Attach the lower end of the shift cable to the cable holder bracket.
8. Install the rear scroll. Tighten the 8 mm bolts to the specified torque (18 ft-lb).
9. If removed, install the wheel covers . Apply grease to the ends of the axle and install the 13 mm washers, ratchet return springs, spacers, keys, and drive gears. Install the 13 mm circlips.
The pinion gears have punch marks on them. Install the left gear so the punch mark faces inside and install the right gear so the mark faces outside.
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Caveat: I work for Honda, but the preceding is my opinion alone.