Fellow up-all-night parents on Twitter were quick to jump in and offer advice to the new dad.
Perhaps the best response was from Bussy McBusface, who explained the word Gayford was looking for was "parenting".
Comedian Jaquie Brown jumped in too, supporting McBusface's complex definition.
Another parent suggested "normal" was the word the Prime Minister's partner might be searching for.
Another, possibly fellow sufferer, labelled the sensation "firstchilditis".
Earlier in the month, Gayford also took to Twitter at 4am while trying to get Neve to nod off. Here he shared how much precision he's realised is required to get a baby to sleep.
"Ah yes of course it was the 2 finger bum tap while rocking just off 180 degrees with light on dim and white noise rolling waves on beach not waves on rocks mind you, and only if turned to the east. That. finally. induced. sleep," he wrote.
"It seems so obvious now as I lay here SO AWAKE."
Read more: • Clarke Gayford's parenting tips: The joy of burping and onesie winners
And in another parenting advice tweet, again at the ungodly hour of 4am, Gayford rated a zip-up onesie against a dome-buttoned number and a double-zipped option. His research led him to deduce the latter was trumps.