ggplot plotting multiple bars

The first thing you want to do is get your data into a proper shape for ggplot2. The philosophy of ggplot is that data is in a long format where you can assign variables of data to different aesthetics, such as color or position, creating your visual elements dynamically. One hint that you probably need to reshape your data is that your columns have very similar names—that's a sign that they contain very similar data.

Think about what it is you want to plot and how you want to put together different elements. If I'm understanding the question correctly, the position along the x-axis depends on the neighborhood and the bracket. The position of the errorbars depend on the neighborhood and the bracket as well. And the endpoints of the errorbars depend on the min and max of the values.

I gathered the data into a long format, and used some regex functions to extract the bracket label and the measurement type (min, max, or neither) from the key column, which contained what were the column names before. Labels where this extract was blank are the measurements themselves, so I filled those in with replace_na, then spread it so there would be a min, max, and measurement for each combination of bracket and neighborhood.

library(tidyverse)

df_tidy <- df %>%
  gather(key = key, value = value, -hood) %>%
  mutate(bracket = str_extract(key, "(?<=PHE_)(\\w+\\s){2}\\w+")) %>%
  mutate(type = str_extract(key, "(min|max)")) %>%
  select(-key) %>%
  replace_na(list(type = "measure")) %>%
  spread(key = type, value = value)

df_tidy
#>   hood       bracket    max measure    min
#> 1    a    20k to 35k 331.15     250 168.85
#> 2    a Less than 20k 368.38     291 213.62
#> 3    b    20k to 35k 185.47     125  64.53
#> 4    b Less than 20k 283.86     220 156.14
#> 5    c    20k to 35k 394.33     306 217.67
#> 6    c Less than 20k 296.82     226 155.18
#> 7    d    20k to 35k 107.94      73  38.06
#> 8    d Less than 20k  41.82      25   8.18

From there, the data is ready to plot, with dodging to put both bars and errorbars side by side. One issue you'll notice is how to fill the bars and color the errorbars: it's hard to see the errorbars where they overlap. One option is to decrease the alpha of the bars.

ggplot(df_tidy, aes(x = hood, y = measure, fill = bracket)) +
  geom_col(position = position_dodge(width = 0.9), alpha = 0.5) +
  geom_errorbar(aes(ymin = min, ymax = max, color = bracket), position = position_dodge(width = 0.9), width = 0.4)

Another is to manually set fills and colors that are similar, but where the errorbars are darker.

ggplot(df_tidy, aes(x = hood, y = measure, fill = bracket)) +
  geom_col(position = position_dodge(width = 0.9)) +
  geom_errorbar(aes(ymin = min, ymax = max, color = bracket), position = position_dodge(width = 0.9), width = 0.4) +
  scale_fill_manual(values = c("skyblue", "tomato")) +
  scale_color_manual(values = c("skyblue4", "tomato4"))

I'll leave those aesthetic decisions to you.