Can I use Time Machine but still be able to access deleted files on my Mac? Normally I just copy important files on my external HDD or move the big files on there cause I don't have much space on my Mac.

However, I want to try Time Machine, because it might be less prone to HDD failure. Since I recently just got a toasted HDD and now all my important files are lost.

Question is, if I want to just move a big file from my Mac to my external HDD just to free up space on my Mac (I'm always low on local storage), can Time Machine do that?
Yes, you can move files away from your Mac to an external HDD when using Time Machine.

The file will remain in the backups until the backup drive runs full and Time Machine starts deleting older backups.

Note however that the backup will contain a version of the file from before you’ve moved it to an external HDD. If you want to backup files on your external hard drive with Time Machine, you’ll need to remove the external drive from the list of exclusions in Time Machine settings.