The plunger is worn, more than I've seen on other guns. It's a very small cost item. You would probably be happier with a new plunger and spring. But more than likely the current one is still functional and just needs a little grease to reduce friction. If the new No.5 sight is wobbly and doesn't stand up straight, then replace the plunger. The plunger won't affect your accuracy. Be sure to put a spot of oil on the retaining pin where it enters and exits the backsight. Remember the adage: "if it rotates, oil it; if it slides, grease it."
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now to buy more enfields now I need a no1 mark III and then a no4 mark 1