In Goblet of Fire, Chapter 2, in Harrys letter to Sirius he wrote
Thanks for your last letter. That bird was enormous; it could hardly get through my window.
Things are the same as usual here. Dudley's diet isn't going too well. My aunt found him smuggling doughnuts into his room yesterday. They told him they'd have to cut his pocket money if he keeps doing it, so he got really angry and chucked his PlayStation out of the window. That's a sort of computer thing you can play games on. Bit stupid really, now he hasn't even got Mega-Mutilation Part Three to take his mind off things.
I'm okay, mainly because the Dursleys are terrified you might turn up and turn them all into bats if I ask you to.
A weird thing happened this morning, though. My scar hurt again. Last time that happened it was because Voldemort was at Hogwarts. But I don't reckon he can be anywhere near me now, can he? Do you know if curse scars sometimes hurt years afterward?
I'll send this with Hedwig when she gets back; she's off hunting at the moment. Say hello to Buckbeak for me.
Why does Harry put this in the letter?
I'll send this with Hedwig when she gets back; she's off hunting at the moment.
Why does Sirius care Hedwig is gone hunting?
It's like saving a draft of an email with the line
I'm saving a draft of this email because I don't have connectivity right now
It can't be to let Sirius know that this was the reason he did not reply back ASAP since in the para before the letter it says :
"Hedwig didn't approve of these flashy visitors" and "reminded Harry to call on him if Harry needed to. Well, he needed to now, all right...".
Also later on Harry says to Uncle Vernon about writing to Sirius:
"It's been a while since heard from me, and, you know, if he doesn't, he might start to think something's wrong."
All this implies Harry had received Sirius's letter a long time ago and he did not reply then, when Hedwig was still there. No point adding it much later when he did decide to write only when his scar hurt and he needed Sirius.