Hold Your Tongue
Kevin Wald
(1) Each of the clues yields a word that has a short form consisting of only a substring of its letters:
AMP[LIFIER]
RHINO[CEROS]
HIPPO[POTAMUS]
[IN]FLU[ENZA]
[BACTERIO]PHAGE
COED[UCATIONAL]
[VIOLON]CELLO
THOU[SAND]
MATH[EMATICS]
[NEIGHBOR]HOOD
KILO[GRAM]
[OMNI]BUS
Using the numbers supplied to index into the remaining letters (the ones that "might or might not be said") yields the following pairs:
FI RS TP AI RI NT OO NS IS BR AG IN
That is, "First pairs in TOON, SIS, BRA, GIN."
(2) Each of those four words is a short form of a longer word ([CAR]TOON, SIS[TER], BRA[SSIERE], [EN]GIN[E]). Taking the first pair in the omitted letters (the ones that are "left out because they are unnecessary") in each word yields CA TE SS EN.
(3) CATESSEN is the omittable part of DELI[CATESSEN]. The answer consists of the letters that have been "left out that are entirely necessary": DELI.