@Gavincenzo Hello, 

How about the 2.4Ghz & 5Ghz Led on RE650? If both LEDs are off, RE650 has not connected to router successfully,  please verify your Wi-Fi password as below and reinstall it.

1. Verify Router's Wi-Fi password : Go to Settings-> Wi-Fi on your phone, Tap (or Long press  the router's WiFi name  ->Forget / Delete the network )
Please reconnect to your router Wi-Fi, and it will ask for a password, will it accept your router password?

2. Reset range extender and follow the instructions on Tether app and reinstall range extender .

Reset: use a pin to press the reset hole for 10 secs while tp-link device is powered on.

If Range extender Leds are On, please let me know the model number of your router, and make sure RE firmware is up to date, let us know the Firmware and hardware of your RE, thank you. 

Model is an RE650 AC2600

Now sometimes my phone connects, sometimes it says the network has no internet connection.

This morning my laptop suddently lost connection to the internet, but was still connected to the network. At this point my phone was connected without issue. Restarted my laptop and it connected without a problem. Went and made a cup of tea, 5 mins later, my phone said the network wasn't connected to the internet, but my laptop was still ok. Seems completely random if/when each device says the network isn't connected to the internet. Same thing happens with my partner's phone.