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btw I won't be in to fire the starting gun, so just start voting at the correct time.

Afraid of being late, he takes hectic short break between two dashes (5,6)

Dash in the sense of "smidgen"; WHIT + BREA(k)* + BIT

A magic prop with bare bit exposed (5,6)

Grace`s song the Rabbi wit composed (5,6)

Caucasian has intercourse with some pig from East End, followed by Alice (5,6)

White
Intercourse - Rabbit and pork (some pig) = talk in Cockney rhyming slang

Just to make sure...

Pig - unpleasant person

Slight tweak....

Caucasian has intercourse with some pig from the East End, followed by Alice (5,6)

Taking my cue from Spike:
A bear and a white rabbit are standing by a river taking a number 2.
The bear says to the rabbit;
“tell me Mr Rabbit; do you find that sh*t sticks to your fur?” The rabbit says “ no, it doesn’t”
So the bear wipes his a*se with the rabbit…..

Pesident's Air Force One to refrain from circling a snowy O'Hare, for one related (5,6)

Jefferson Airplane
Refrain - song
(O)Hare
Related - Appropriate or suitable in the circumstances

typo in 19, spike - pResident