I have a paper wallet and the private key. What methods can I do on the desktop to send a portion of these bitcoins using only a desktop app? Is there an app where I can type in my private key and select how much to send?
When you launch Electrum for the first time, it wants you to create a new wallet. Same when you just choose
File → New/Restore at any point.
Then, choose "Restore a wallet or import keys" and "Standard wallet". On the next page, enter your private key. The password you can choose afterwards is optional and you can skip it or only remember for the period of time you're keeping this wallet on your computer.
If you want to try this in advance without typing your own private key in, you can use the private key
KzbvEHb1HvDRZMPvVF7Du9F4N5g22QkeoeZ7qPCkjSPktt5bbCDU. Is has had 50 mBTC going in and out again as you can see on the blockchain.info page about its corresponding address
15dNGcjVE8ddCt3vgYwcpJjYXK4ydZtykF and you can find out how Electrum shows this to you if you create a wallet using it. This private key has been specifically created by me to be publicly known.
Note that Electrum won't be able to create more addresses from it because it only is a single private key. It is, however, able to show the one address the private key corresponds to.
You can use my answer to your question " How do you send to multiple addresses by paying a one time fee? Which wallet supports this?" to transfer the remaining money back to the address it came from so you can spend it using the same private key in the future.
Most desktop wallets will let you do an import of a private key. For security reasons, they usually will make you spend the amount to another address.
Armory has a private key import feature and you can spend some back to the same address (not recommended).