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How to add a second set of labels to second set of data points
I have added labels (2016 response rate) to the bars of my barchart but want to also add labels to the individual/black data points (2015 response rate) that lie inside each bar.

This is my code:
ggplot(merged2[merged2$TrustCode %in% acuteCodes, ], aes(x = TrustName, y = ResponseRate16)) + 
  geom_bar(fill="white",stat="identity", colour="blue") +
  geom_point(aes(x = TrustName, y = ResponseRate15), shape=18, size=3, colour="black") +
  geom_text(aes(label=ResponseRate16, x=TrustName, y=1.10*ResponseRate16), colour="black")

How can this be achieved?

Exactly like you mapped the bar values: add a geom_text()
ggplot(merged2[merged2$TrustCode %in% acuteCodes, ], aes(x = TrustName, y = ResponseRate16)) + 
  geom_bar(fill = "white", stat = "identity", colour = "blue") +
  geom_point(aes(y = ResponseRate15), shape = 18, size = 3, colour = "black") +
  geom_text(aes(label = ResponseRate16, y = 1.10*ResponseRate16), colour="black") +
  geom_text(aes(label = ResponseRate15, y = 1.10*ResponseRate15), colour="red")

I slightly cleaned up you code of repeated x mappings. No need to duplicate an aesthetic if it's already mapped in the initial ggplot function.
You probably also want to use the nudge_y arg instead of setting a new y aesthetic.
ggplot(merged2[merged2$TrustCode %in% acuteCodes, ], aes(x = TrustName, y = ResponseRate16)) + 
      geom_bar(fill = "white", stat = "identity", colour = "blue") +
      geom_text(aes(label = ResponseRate16), nudge_y = .5, colour="black") +
      geom_point(aes(y = ResponseRate15), shape = 18, size = 3, colour = "black") +
      geom_text(aes(label = ResponseRate15, y = ResponseRate15), nudge_y = .5, colour="red")


You know, I tried this and clearly must have typed it wrong (because it didn't work), so I presumed that this wasn't the way! I think it's time I took a break.  (And thanks for the cleanup!)  Whilst you're here, is there a way to display the black numbers to 2 decimal places and also add percentages to all numbers?

You probably also want to use the nudge_y arg instead of setting a new y aesthetic. Editing answer for that
