@Wayne-TP 

Thank you for the answer.

I don't have any VPN or ad-blocking software on my phone. I had only AVG antivirus, but I uninstalled even that one. My SIM card signal is very good and looks quite stable, and my home internet is around 500 Mbps download and 250 Mbps upload. I tried to use my phone with other Wi-fi networks, away from home, with the same results. There is the same delay, mostly around 2 min, sometimes more, sometimes less, sometimes I have to restart the app to make it display the "Energy Usage". My phone is quite new, with 8 GB RAM and 256 Mb storage, mostly empty, and works fine with all the other apps. In the background there was only the AVG protection running, but even without it there is no change.

  @Wayne-TP 

Thank you again for the answer.
I updated this replay after some new findings. So, my current solution that seems to work fine until now was to change the router settings to this:
- wireless frequency - "2.4 GHz" and "5 GHz" (is the default setting but seems to not change anything),
- wireless mode - "802.11n"  (802.11g/n mixed" and "802.11b/g/n mixed" settings seems to be the main cause of the problem... I even reset the router to the default settings to check it and I encountered the same problem until I changed the default "802.11g/n mixed" setting to "802.11n"),
- channel - "manual" to "channel 11",
- channel width - "20MHz" (is the default setting),
- security - "WPA2-PSK" (is the default setting),
- IPv6 firewall protection "enabled" (is the default setting), but with IP flood detection and ICMP flood detection "disabled".

PS. I will keep testing this settings a few days before checking this solution as the "recommended solution".

Update: There are still some random inexplicable delays between 15 seconds and a few minutes, but only occasionally.

Update 2: When I opened the app in the morning, I again had a delay of around 2 minutes until I got the  "Energy Usage" displayed and acquired control over the plugs. It looks like whenever I don't acces the app for a while, the old behavior comes back.

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My app version: TP-Link Tapo 3.2.976

Country: Romania

SIM card service provider: DIGI Mobil

I didn't tried on other mobile devices yet.

All the problems seem to be still there with delays from 30sec to a few minutes. It works for a while and then, randomly, it starts to have delays again, then it works again for a while, and again stops working. This behavior is the same, no matter if I use my mobile internet, my home WI-FI network or some other WI-FI network. I had some expectations from a TP-Link product, but... 

Anyway, I don't know if this is a hardware or software problem, but I will return the product to the seller and try another brand because I have already wasted too much time with something that should have been made to work fine in a matter of minutes.

Update: I returned both plugs and bought a smart plug from a different brand/producer. I tested it for a few days, and it worked like a charm (even with the router on default settings), with the dedicated app displaying the energy info and responding always under 1 sec, even 2 rooms and 1 floor away from the router. Because both Tapo P110 had the same problematic behavior, it means there is a problem with all these types of plugs (Tapo P110) or with the app that controlled both the plugs.