Correct use of 'whatsoever'
Solution 1:
The word whatsoever is a synonym for the phrase at all in these sentences. The following two formulations are equivalent:
The governor may issue pardons for any reason whatsoever.
The governor may issue pardons for any reason at all.
The meaning indicated is that there is no restriction on what the governor may do. The words whatsoever and at all both intensify the word any.
Note that whatsoever is NOT a synonym of whatever in this context, as the word whatever cannot be used in the sentence above. However, in an archaic mode one may use whatsoever as a synonym of whatever when the word is being used as a pronoun:
Whatever you do, don't fall down.
Whatsoever thou doest, fall not down.