read through the incomming HTTP-Header fields and store that information.
On the next request - add this to your next outgoing message (as additional header field). I'm not shure about the proper naming convention, but this should be findable via Google...
edit:
ok should be straight forward: HTTP cookie - Wikipedia
you receive several "Set-Cookie" and concatenate them in your next request as "Cookie".
edit 2:
well, that's seems to work. If you send the received cookie, at least the Arduino test page from the webclient example doesn't send the __cfduid again:
22:03:10.503 -> try www.arduino.cc connected IP 172.67.5.5
22:03:10.863 ->
22:03:10.863 -> client connected
22:03:10.863 -> HTTP/1.1 200 OK
22:03:10.863 -> status code=200
22:03:10.863 -> reason phrase=OK
22:03:10.863 -> Date: Tue, 27 Oct 2020 21:03:12 GMT
22:03:10.863 -> Content-Type: text/plain
22:03:10.903 -> Content-Length: 6
22:03:10.903 -> Connection: close
22:03:10.903 -> Set-Cookie: __cfduid=d93b605c6df40be709439f9f96cf63b041603832592; expires=Thu, 26-Nov-20 21:03:12 GMT; path=/; domain=.arduino.cc; HttpOnly; SameSite=Lax
22:03:10.903 -> D162 Set-Cookie found
22:03:10.903 -> Last-Modified: Mon, 31 Aug 2020 09:25:41 GMT
22:03:10.903 -> ETag: "5f4cc215-6"
22:03:10.903 -> Accept-Ranges: bytes
22:03:10.903 -> CF-Cache-Status: DYNAMIC
22:03:10.903 -> cf-request-id: 060d7850370000cbb4dd8f6000000001
22:03:10.903 -> Server: cloudflare
22:03:10.903 -> CF-RAY: 5e8f5cc6afc7cbb4-VIE
22:03:10.903 ->
22:03:10.903 -> 10813
22:03:10.903 ->
22:03:10.903 -> body=
22:03:10.943 -> success
22:03:10.943 -> 0
22:03:10.943 -> 0.00
22:03:10.943 -> ⸮
22:03:10.943 -> Received 494 bytes in 0.3000ms, rate = 1.61 KB/sec
22:03:40.749 -> try www.arduino.cc connected IP 104.22.49.75
22:03:41.199 ->
22:03:41.199 -> client connected
22:03:41.199 -> HTTP/1.1 200 OK
22:03:41.199 -> status code=200
22:03:41.199 -> reason phrase=OK
22:03:41.199 -> Date: Tue, 27 Oct 2020 21:03:42 GMT
22:03:41.199 -> Content-Type: text/plain
22:03:41.199 -> Content-Length: 6
22:03:41.199 -> Connection: close
22:03:41.199 -> Last-Modified: Mon, 31 Aug 2020 09:25:41 GMT
22:03:41.199 -> ETag: "5f4cc215-6"
22:03:41.239 -> Accept-Ranges: bytes
22:03:41.239 -> CF-Cache-Status: DYNAMIC
22:03:41.239 -> cf-request-id: 060d78c64e00000d6398890000000001
22:03:41.239 -> Server: cloudflare
22:03:41.239 -> CF-RAY: 5e8f5d838ef70d63-VIE
22:03:41.239 ->
22:03:41.239 -> 10813
22:03:41.239 ->
22:03:41.239 -> body=
22:03:41.239 -> success
22:03:41.239 -> 0
22:03:41.239 -> 0.00
22:03:41.239 -> ⸮
22:03:41.279 -> Received 339 bytes in 0.3830ms, rate = 0.86 KB/sec
After a reset, when the Arduino hasn't a stored cookie - Arduino.cc will send a new cookie.
But it takes a lot of RAM (Arduino sends a 53 byte cookie), so you have to take care that you have enough global RAM available to store such long cookies.
I can't post the code today because it needs some cleanup and a malicious server could hang the client, but basically ... yes it can be done.