From what I know, TypeApp stores your messages on their servers. So, their server accesses your mail first from Yahoo server, and then you client receives it from TypeApp's server. Thus, Yahoo in that case doesn't distinguish what network you are on: it doesn't know.
Now the question is whether you want your messages to be stored (and analyzed) on a 3rd party's server...
I understand that ultimately, you are looking for a working solution. Personally, I would have abandoned Yahoo Mail long time ago. (I've never really used it myself, - only for something unimportant. But several relatives and friends have done that.)
Note, that you can (1) make the switch gradual, - check how you like a new provider. (Personally, among free mail providers, I'd consider Gmail, maybe GMX.) If you do, then either establish a forward from Yahoo, or configure the new provider collecting mail from Yahoo (E.g. Gmail can collect mail from other accounts via POP3), so that whoever has your old (Yahoo) address can still reach you.