2002 Honda civic

I have a 2002 Honda civic (manual). Lately, when I have pushed in the clutch to start it, the engine makes a bad sound. It sounds similar to the sound an automatic would make when you turn the key while it is already running. Does anyone know what that might be?

Let me figure out exactly what you mean…and when it happens…

Engine off,
Push in the clutch, - here?
Turn the key - or here?
Or after all that? Or something else altogether?

Don’t an automatic and stick sound the same, when you turn the key to start it when it is already running?
Do you mean that it makes a weird whining sound when you’re starting, fast and screetchy?

Not sure what you mean by “pushed the clutch in while starting”. Of course, you can start a car in neutral but it is always a good habit to press the clutch while starting, just in case.
Or are you suggesting that the noise does not occur when you’re starting in neutral, without pressing the clutch?

Have the starter removed and then inspect the condition of the starter driver gear teeth and the condition of the teeth on the flywheel.

If the starter drive gear teeth are damaged replace the starter. If the teeth on the flywheel are damaged the transaxle will have to be removed to replace the flywheel.