I finally finished the* Cursed Child*…it seems that reading five things at once slowed me down a bit.
The general opinion of the script seems to be negative, although I personally enjoyed it. It definitely had its flaws, and there were some inconsistencies between it and the books. Warning: there are some major spoilers below.
I found Albus and Scorpius’ interactions to be odd at times. For the majority of the play, they were supposed to be fourteen, but there were times when their speech didn’t reflect how teenagers typically speak. Here, Scorpius in particular acts more like a goofy little kid than a fourteen-year-old.
Ron, too, I thought was shortchanged. He came across as little more than comic relief, and in one scene, he even charged into a room with his wand pointed the wrong way. He may not have been the brains of the group but he was never quite that stupid.
Also, the way it depicts Time Turning is at odds to how it is portrayed in
Prisoner of Azkaban. If you use a Time Turner to go back in time, you have to live through that time again. Time Turners don’t bring someone back to the present. So even though these Time Turners are able to go back years instead of merely an hour, the script doesn’t explain why the time traveller is able to return to the present almost instantly.
Also, the ease with which they brew Polyjuice Potion is strange. In a latter part of the play, the characters suggest sneaking into Bathilda Bagshot’s house and stealing Polyjuice ingredients so that Harry could transform into Voldemort. He ends up using Transfiguration to transform into him instead (though that too is strange…this renders Polyjuice Potion a bit redundant). But what they apparently forgot is that Polyjuice Potion is supposed to take six months to prepare.
The major issue to me though is the idea of Voldemort and Bellatrix having a daughter. Voldemort believed himself to be immortal due to the Horcruxes, so I can’t see him finding it necessary to produce an heir. Plus Voldemort seemed more…wraith-like to me than actually human. The idea of someone, even someone as fanatically devoted to Voldemort as Bellatrix, sleeping with him just seems…odd. I didn’t mind Delphi as a villain
per se, or her character and motivations. Her origins are just too contrived as Robyn p pointed out.
So yes, there are a fair number of plot holes and inconsistencies in the story. But I still found myself enjoying it. I was glad to read about these characters again.
Overall, I liked the themes. Harry’s struggles to be a good father to Albus and their eventual reconciliation was portrayed well IMO. His issues are realistic given that he never really had a father figure growing up. There were some touching scenes as well, especially when Harry spoke to Dumbledore’s portrait about his fears and lingering bitterness.
I didn’t mind the time travel aspect (beyond the logic issues with the Time Turners) especially as we had the chance to encounter some other characters from the books. Those cameos were a pleasant surprise, with the exception of Umbridge.
It may be pandering to nostalgia and fans (hence the accusation that the play is too much like fan fiction), but admittedly I fell for it.

I just wish there was more Hagrid!
I think part of the problem was the format. It’s a play, it’s meant to be performed, not read like a novel.
I agree…this is best experienced as a play. There are certain scenes that would be interesting to see in person, e.g. the living bookcase that eats people, the Dementors rising from the audience, the wand duels. Those who have actually seen the play have a more positive opinion of it, and that may be due to the production value and special effects. I would like to see it performed.