If you're looking for the single mean for just the weekdays, you could do something like this:
dat = data.frame(Time = rep(c("00:00","01:00"),c(7,3)),
                 Day = c("Sun","Mon","Tue","Wed","Thu","Fri","Sat","Sun","Mon","Tue"),
                 Count = c(169,71,70,68,91,94,135,111,45,50),
                 Speed = c(60.2,58.5,57.2,58.5,58.8,58.7,58.5,60.0,59.2,57.6))
mean(dat$Count[dat$Day %in% c("Mon","Tue","Wed","Thu","Fri")])
# [1] 69.85714
If, on the other hand, you're looking for the mean across each individual day then you could do this using base R:
aggregate(dat$Count, by=list(dat$Day), FUN = mean)
#   Group.1   x
# 1     Fri  94
# 2     Mon  58
# 3     Sat 135
# 4     Sun 140
# 5     Thu  91
# 6     Tue  60
# 7     Wed  68
It looks like you've tried dplyr, so the syntax for that same operation in dplyr would be:
library(dplyr)
dat %>% group_by(Day) %>% summarize(mean_count = mean(Count))
#  Day   mean_count
#  <chr>      <dbl>
# 1 Fri           94
# 2 Mon           58
# 3 Sat          135
# 4 Sun          140
# 5 Thu           91
# 6 Tue           60
# 7 Wed           68
And if you want to do the same thing in data.table you would do this:
library(data.table)
as.data.table(dat)[,.(mean_count = mean(Count)), by = Day]
#    Day mean_count
# 1: Sun        140
# 2: Mon         58
# 3: Tue         60
# 4: Wed         68
# 5: Thu         91
# 6: Fri         94
# 7: Sat        135