How to increase space between the given two stacked bars?

You could change the xmin of your third bar and xmax of your second bar using ggplot_build. I changed it with 0.1 so you can modify that. Here is a reproducible example:

library(ggplot2)
library(dplyr)
Year      <- c(rep(c("2006-07", "2007-08", "2008-09", "2009-10"), each = 4))
Category  <- c(rep(c("A", "B", "C", "D"), times = 4))
Frequency <- c(168, 259, 226, 340, 216, 431, 319, 368, 423, 645, 234, 685, 166, 467, 274, 251)
Data      <- data.frame(Year, Category, Frequency)

p <- ggplot(Data, aes(x = Year, y = Frequency, fill = Category, label = Frequency)) +
  geom_bar(stat = "identity") +
  geom_text(size = 3, position = position_stack(vjust = 0.5)) 

# Old plot
p

q <- ggplot_build(p)

# Here you can decide the distance between specific bars
q$data[[1]] <- q$data[[1]] %>%
  mutate(xmin = ifelse(x == 3, xmin + 0.1, xmin),
         xmax = ifelse(x == 2, xmax - 0.1, xmax))

q <- ggplot_gtable(q)
# New plot
plot(q)

^{Created on 2022-08-26 with reprex v2.0.2}

Here is a quick hack, but colleagues with more expertise will no doubt provide more elegant answers:

1. rename Year to something else, and add a new continuous Year that represents the gaps you want:

Data <- Data %>% rename(grp=Year) %>% mutate(Year = rep(c(1,2,4,5),each=4))

2. add scale_x_continuous() indicating the breaks and labels you want.

scale_x_continuous(breaks=c(1,2,4,5),labels = unique(Data$grp))