I think (though could be very wrong) that they provide streaming of all that they have the rights for. So, I would expect them to provide some of the Aussie Open (it'll be one of the outside courts though). I don't know who has the tv rights for Sydney or Auckland, or Zagreb so don't know about that.
There is live streaming of ATP Auckland from the Auckland website, however it's a paying thing.
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Can't tell you if Eurosport Player has whole matches on afterwards - they have had nothing to show for ages, so nothiing available right now. Prime though - plenty of choice - loads of WTA matches going back to 8th January.
So I hope Prime get the AO next year as well.
Eurosport are doing a deal at 99p for one month if you sign up anytime before 21/2/21 i.e. valid for the AO.
Thereafter it's £6.99 pcm so you'd need to cancel the free offer before your month is up ...
This was noted on Twitter by Stu Fraser and appears to be a pukka offer - but read the small print.
I had it last year and forgot to cancel so will keep it going for the AO coverage but haven't really used it much to be able to answer your queries. And I'll probably cancel it after the AO anyway.