I decided to watch this simply for the cover art, it gave me the feeling that this might be a fun watch. Though the interesting sounding synopsis did help.
I'm sure I'm not the only one who picks anime this way, it's not the most ideal way, but sometimes it has great outcomes.
Maybe I have terrible taste, or maybe other people judge harshly, either way, I actually really enjoyed Tokyo ESP!
I've seen people say the art is terrible. It's different, but it's not terrible.
Also that the story is jumpy and fast, it's not. Everything works its way together. If you pay proper attention you're fine.
There are some things that aren't explained, but if the series continues (which the ending is left open with a continuation insinuated), they're the type of things that probably will be explained then.
As soon as it starts, Tokyo is being attacked by a group of Espers.
Then the White Girl turns up.
Kyoutarou wants to protect the world from Espers doing bad things, and he wants Rinka to help him.
The plot starts coming together and new characters start coming in.
Rinka's dad, Rindou, he's giant, but also a softie.
Kobushi, she's usually trouble, but comes more involved toward the middle.
Murasaki, a young girl that Rinka and Kyoutarou make friends with.
The group of enemy Espers are killing, stealing and destroying whatever they like, for what reason it's not made clear at first.
It is a darker styled anime, with a heavy focus on powers and random phenomenon, but it has it's comedic and even romantic points too.
If it sounds interesting to you, give it a chance, don't let the low rating put you off something you might really enjoy, like I did.