facet_wrap add geom_hline

You could also create your own stat to calculate the line for you. Adapting the example from the extending ggplot2 guide You can make

StatMeanLine <- ggproto("StatMeanLine", Stat,
  compute_group = function(data, scales) {
    transform(data, yintercept=mean(y))
  },
  required_aes = c("x", "y")
)

stat_mean_line <- function(mapping = NULL, data = NULL, geom = "hline",
                       position = "identity", na.rm = FALSE, show.legend = NA, 
                       inherit.aes = TRUE, ...) {
  layer(
    stat = StatMeanLine, data = data, mapping = mapping, geom = geom, 
    position = position, show.legend = show.legend, inherit.aes = inherit.aes,
    params = list(na.rm = na.rm, ...)
  )
}

Then you could use it like

ggplot(mtcars, aes(mpg, cyl)) +
  stat_mean_line(color="red") +
  geom_point() +
  facet_wrap(~ gear)

Minimal example using mtcars - you have to create a data frame with mean for each gear (in your case it's Name).

library(tidyverse)
dMean <- mtcars %>%
    group_by(gear) %>%
    summarise(MN = mean(cyl))
ggplot(mtcars) +
    geom_point(aes(mpg, cyl)) +
    geom_hline(data = dMean, aes(yintercept = MN)) +
    facet_wrap(~ gear)

For your case this should work:

library(tidyverse)
dMean <- UKWinners %>%
    group_by(Name) %>%
    summarise(MN = mean(TE.Contr.))
ggplot(UKWinners) +
    geom_point(aes(Pcode, TE.Contr.)) +
    geom_hline(data = dMean, aes(yintercept = MN)) +
    facet_wrap(~ Name)