I'm expert on that as I'm also a singer and I do recording stuff. You need to use some sort of "noise gate" or "expander" for that (noise gate completely cut off noise below a certain DB threshold, whereas expander just reduces it gracefully). What I did in my first course was recording the voice on a separate audio file (separate from the video) using a Rode NT2 microphone, which is a professional studio mic used to record singing voice. I used Logic Pro as software and then I created a channel strip with different settings. The secret is the Multiprocessor plugin, where, in the expander section, you can configure the threshold that you need. You will also need some compression, equalization and reverb. In the end, it should sound stunning, like a podcast. Also, note that you should avoid noise that normally you cannot hear but affects the recording. When I sing and I put the headset on and there is no backing track playing I can hear any far noise amplified, so that tells me there are a lot of noises around us. Just turn off your fridge and any electrical device that might affect your recording. I hope this helps. Note that the fact there is no noise also allows you to record voice in multiple "takes" like singers normally do for songs (one line at a time), and also allows you to edit single words that you mispronounced.
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