Attackers knows all about automation. Ten seconds are a lot of time for an attacker who knows how to write scripts. In your scenario, if the insecure PC is malicious, it can use this very small window to enact a lot of harm: contrary to a human user, the attacker is not limited to how fast he can move a mouse.
I suppose that you want to login with your smartphone, and, from that smartphone, allow access to your account for "ten seconds" from the possibly malicious PC; the idea being that you do not want to type your precious password on that PC. This will indeed be beneficial to you if you assume that the PC just passively grabs passwords, but will not actively connect to whatever Web site you wish to use. Not all attackers can be trusted for being so polite, though.
Another similar but much more common and robust solution is to use one-time passwords. Instead of opening your accounts to everybody, even for a short time, use a password which works only once. Key loggers can log it, it won't matter, since the server will not accept each password more than once. You still have all the issues with maliciously active insecure PC: once you accept to use the bad guy's machine, you are giving him, virtually, access to your account for at least the few seconds you are using it, and that's enough to forfeit your security.