DIAGNOSIS That's the sound of the paper pickup rollers slipping or a foreign object in the paper feed path. The rubber rollers get smooth and glossy over time and lose their grip.
FIX Power off and unplug. Open the front cover and remove the paper tray. You'll see a set of rubber rollers at the bottom of the paper feed area. Dampen a lint-free cloth with a little bit of warm water (not alcohol or cleaner, which can dry the rubber out more). Wipe the rollers thoroughly to remove any paper dust and restore tack. Let them dry completely before plugging the printer back in.
IF THAT DIDN'T WORK The noise might be coming from the separation pad, a small rubber piece that helps feed one sheet at a time. It's located right where the paper first enters the machine. Check for any worn spots or debris on it. Sometimes a small piece of label or a staple fragment gets lodged in there and causes the click.
HEADS UP If cleaning doesn't stop the noise and the slipping, the pickup roller assembly (part # PK- NV) is a replaceable part. It's a known wear item on these Brother models.