I have the private key backup from my Android phone wallet. My phone broke and has been replaced but I can't see any import option in the Android wallet app.
How do I get my BTC back?
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Since I don't know what wallet app on Android you're using, I'll give an answer that should work for any:
If you create an account at Coinbase, you can import your private keys and then send them to an address on your Android wallet. This requires the private keys being in plain text. If your private key export is encrypted (with a password you gave it) or encoded in some non-obvious fashion, you'll need something to decode it before using this option.
If it's encrypted in a way that's compatible with the desktop Bitcoin-QT client, you can reclaim it that way.
If it's encrypted some other way, you'll likely need to restore it to your Android phone and enter a password there to get access to your bitcoins. This would be specific to the app you're using.
If you're using Bitcoin Wallet, which seems to be the closest thing to an "official" Android client, choose "Back up Keys" from the menu (on my device, I press the Menu hardkey to open that), and then click "Restore private keys".