what is the best way to change the colour of an image do I overpay the colour I want?

If I delete the red colour all the detail of the image will disappear. I need to change the colour of the image from red to whatever cmyk or spot colour I or (the client wants)

Pick the red color using the color picker:

Open the layer by clicking the lock in the right:

Press delete to delete the red color:

Remove the selection Ctrl+D and make layer below the existing one:

select color you need:

Make the layer in the top in white or with the color you need: Cmd+Shift+Backspace
or convert it to grayscale using cmd+shift+U

or like that: