For a start, according to the device list for this programmer at
https://www.electro-tech-online.com/custompdfs/2010/05/864_devicelist.pdf, the PIC16F57 is not one of the devices it can reprogram. Also. looking at the pictures of this programmer, you would have to take the chip off the board to place it into the programmer. I trust it has been socketed on the board!
However, if the board has a 6 pin plug on it with the labels you mentioned in your original posting, then you should be able to program it while it is still in place (that's the "In Circuit" part of the ICSP name). The programmer has a 6 pin cable on it that will connect directly into the plug on the board. The programmer then connects to your computer and the software in that will communicate with the programmer and upload the binary file to the PIC device.
I would suggest that you either spend the $40 odd dollars for a PICKIT programmer (don't quote me on the prices - I doubt if we are even in the same country!) or, as you are a school, you might be able to persuade a nearby University or commercial outfit to let you use one of theirs.
Having said all that, I have an uneasy feeling about what you are trying to do. Do you have the source of the program that is currently in the PIC device? What programs are you using to edit and compile the source? Are you able to simulate/test your program changes before you upload them into the chip, or are you planning on using the debugger capabilities of the programmer to fix any problems you encounter?
If you are using the free software from Microchip, then it should be able to connect to and communicate with the PICKIT programmer/debugger without a problem. If you are using something else, then you will need to make sure that it operates correctly with whatever programmer you are able to get access to.
Also, if you are needing to do any debugging by interacting with the programmed board itself, then you may need something other than a bit-banging programmer as they tend to only perform the programming function.
Susan