NavList:

Frank, seems to be a three-persons-observation, making simultaneous observations of lunar distance, sun LL altitude and moon UL altitude. Or perhaps four persons if there was a dedicated chronometer reader. In the four columns to the left are chronometer reading, distance, sun altitude, moon altitude. The three top rows shows the three sets of observations; on the fourth row their sum and on the 5th row their mean values. 45″ have been added to the distance, this seems to be an instrument correction; also 16′15″ for sun sd and 15′57″ for moon augmented sd. Altitude corrections +12′ for the sun and -20′ for the moon as you described. A chronometer error of 22^m36.4^s fast. This ends up in the following: chronometer time 0^h36^m36^s, apparent lunar distance 90°44′57″, sun app alt 65°45′, moon app alt 23°21′.

The clearing process is according to the "second method" as described in Bowditch of 1851. "To the proportional logarithm of the moon's horizontal parallax, (Table XXII.) add the log. cosecant of the star's apparent altitude, (Table XXVII.) the log. sine of the star's apparent distance, ...". The sum of these logs are shown below the sun and moon altitudes, .5308 and 2.7756 respectively. Below the apparent lunar distance are shown the 1^st, 2^nd and 3^rd corrections, which are added to the distance, and then 10° is subtracted from the sum. The 10° is not shown but included. The cleared distance is 89°54′15″. The noon lunar distance from the almanac is 90°11′46″ with its associated proportional log 0.2783 (it looks like 0.2789 is written but 0.2783 makes the sum correct and seems to better describe the rate of change in the distance). The difference between cleared distance and almanac distance is 17′31″ and its plog is 1.0118. The plog difference of 0.7335 gives GMT 33^m15^s which is used in the leftmost column to determine the chronometer error, 3^m21^s faster than previously believed; and also used to determine longitude by lunar by taking the difference between GMT and LMT, 2^h24^m07^s or 36°01′45″.

LMT is determined by an ordinary time sight based on the sun's altitude, shown in two columns at the right. The latitude is 17°40′ south, polar distance 67°41′, equation of time 9^m24^s. Here the longitude is calculated using the original chronometer error and becomes 36°52′. 

I've seen worse handwriting.
Lars