Hello  @Mor1204 

• If it's a Guest Wi-Fi network of the router, the range extender could connect to that guest Wi-Fi network to extend Wi-Fi range. 
• But from your description that Wi-Fi  is also Portal Wi-Fi (authentication page pops and requires login to use internet), as public/ portal Wi-Fi has different ways for authentication, range extender may have trouble connecting to a portal Wi-Fi network, and we can't guarantee that RE will work in such condition. Before purchasing, it's suggested to double-check with portal/ public wifi support whether that portal Wi-Fi could be extended by extender.

If you already got tp-link range extender, you could try to set up range extender and see if that work.

TP-Link Range Extender Setup (Range Extender Mode)