Once a hurricane is identified, it is given a name from a list drawn up by the United States Weather Service, a list that is reused after a few years,
but with the names of the worst hurricanes omitted.
A.
but with the names of the worst hurricanes omitted -- this is somewhat a modifier for "list". The list will be used after a few years, the sentence is completed here but the underlined part just describes the "list" quality that the "list" doesn't contain worst hurricanes names. This is correct
B.
omitting the names of the worst hurricanes -- It changes the meaning. The list will be used when omitting of the names is happening.
C. the names of the worst hurricanes
being omitted -- "
being" makes it awkward.
D. after
they have omitted the names of the worst hurricanes -- "
they" doesn't have a proper referent.
E.
after omitting the worst hurricane names -- it says the list will be used after omitting the names is done. Though this action may make sense, compared with A this is not appropriate as here another action is included before using the list after a few years. This is not the intended meaning.
OPTION:
A