Flight Instructor reported the student lost control of the aircraft on landing and the instructor was unable to prevent the runway excursion and propeller damage.

Narrative

Student pilot training flight for traffic pattern operations and TOLs (takeoffs and landings). Student flying is a new student pilot with 8 hours going into the incident flight. In the previous three flights; the student pilot practiced the standard traffic pattern; and received ground instruction on traffic pattern operations. In the most previous flight; the student pilot expressed that they would like the instructor to start relinquishing the controls more often; and to let them partly get into a mistake and fixing the problems on their own. The instructor agreed to start giving the student pilot more control. The student pilot was at the controls; and the instructor gave commands for the students to follow. On final; the instructor gave pointers to the student to maintain runway centerline using the rudders. Upon touchdown; the plane landed in a side-loaded condition; and the instructor commanded the student to get centerline back using the rudders. At this point; the student tried to get on centerline; but fishtailed and lost control in the process. Upon entry of the fishtail; the instructor took full control of the aircraft; but was unable to establish positive control. The plane went onto the grass; roughly five feet right of RWY XXL. The instructor did not initiate a go-around due to not having positive control of the aircraft and not wanting to aggravate the situation.The aircraft slowed itself down short of Taxiway 1 on the grass; and was promptly taxied onto Taxiway 1 to assess the situation. ZZZ Tower asked if they required any assistance. The instructor deemed the damage to be non-intrusive for taxi operations; so they requested to taxi back to the ramp. Upon shutdown; the instructor stepped out of the plane for a full inspection and found that the edge of one side of the propeller was bent about twenty degrees back . Both parties left the plane with no bodily injuries.