How can I repair a deep gouge on my hardwood floor?

If you have access to saw dust from similar colored wood or find a stain color that matches mix a small amount with sawdust to get color to closely match. Once you have desired color to match let dry once it is dry mix with wood glue to fill in the gouge pushing down as you do so. When it is filled and dry you will need to very lightly sand it a bit to get it to be at same level as the floor be very careful to not over sand. When I did this to our floor I used a tiny bit of polyurethane to cover & seal it also lightly sanding the poly once it was dry so it would match the rest of our floors shean.

They make a hard wax sticks in various wood colors to fill gouges... https://www.youtube.com/watch?v=yJap02hidHE

or wood filler.... https://www.youtube.com/watch?v=cOw6v6Q9I-4&t=53s

https://www.youtube.com/watch?v=eXWWslWF8V4

Go to www.fastcap.com and order the SoftWax kit. There will be a color to match your wood. Rub it on like a crayon, level it off and buff it out.

fill with wood filler putty

If it's really deep may need wood filler than use the magic marker type pens that are for cover wood..can find at Lowe's

Use Bondo wood putty. It works well. You can stain it to match your current floor. I could not find it in the stores, so I had to order it online from Amazon.