You don't describe your computer system. Without that info, it's difficult to answer your query.
If you are running Windows (any version), the RECORDING CONTROL window (accessible via your Control Panel - Sound & Audio Devices) should include a slider for STEREO MIX.
Click on Stereo Mix to make recordings using AUDACITY (or any recording software) of all active input devices (including your microphone and any receive audio from the HamSphere (HS) transceiver.
Be aware that your TRANSMIT AUDIO LEVEL will also be affected by the setting of the STEREO MIX slider. You will have to experiment to find a setting of the STEREO MIX slider that will yield a good recording AS WELL AS proper level transmit audio. You may have to adjust the MIC LEVEL on the HS transceiver in order to obtain a good clean level on transmit that will also record properly.
Everyone's setup is a little different. Be prepared to experiment a bit to find the best settings for your setup.
Also consider using a higher end recording software package. AUDACITY is free and works fairly well. Better packages are offered by Adobe, Sony and Blaze Audio that will give you more flexibility, adjustment range and even effects. The higher end packages do cost money, of course. I've had great success with Sony's Sound Forge software which I use to make professional quality voiceover recordings as well as recordings off HS.