add new column represent the number of occurrence of weekday within the specific month

This is a way to do it

library(dplyr)
library(lubridate)

# Sample data in a copy/paste friendly format, use your own data frame instead
sample_df <- data.frame(
  stringsAsFactors = FALSE,
              date = c("11/01/2022","11/02/2022",
                       "11/03/2022","11/04/2022","11/05/2022","11/06/2022",
                       "11/07/2022","11/08/2022","11/09/2022","11/10/2022",
                       "11/11/2022","11/12/2022","11/13/2022","11/14/2022",
                       "11/15/2022","11/16/2022","11/17/2022","11/18/2022",
                       "11/19/2022","11/20/2022","11/21/2022","11/22/2022"),
          day_name = c("Tuesday","Wednesday",
                       "Thursday","Friday","Saturday","Sunday","Monday","Tuesday",
                       "Wednesday","Thursday","Friday","Saturday","Sunday",
                       "Monday","Tuesday","Wednesday","Thursday","Friday",
                       "Saturday","Sunday","Monday","Tuesday")
)

sample_df %>% 
    mutate(month = month(mdy(date))) %>% 
    group_by(month, day_name) %>% 
    mutate(weekday_occurrence = row_number())
#> # A tibble: 22 × 4
#> # Groups:   month, day_name [7]
#>    date       day_name  month weekday_occurrence
#>    <chr>      <chr>     <dbl>              <int>
#>  1 11/01/2022 Tuesday      11                  1
#>  2 11/02/2022 Wednesday    11                  1
#>  3 11/03/2022 Thursday     11                  1
#>  4 11/04/2022 Friday       11                  1
#>  5 11/05/2022 Saturday     11                  1
#>  6 11/06/2022 Sunday       11                  1
#>  7 11/07/2022 Monday       11                  1
#>  8 11/08/2022 Tuesday      11                  2
#>  9 11/09/2022 Wednesday    11                  2
#> 10 11/10/2022 Thursday     11                  2
#> # … with 12 more rows

Created on 2022-12-21 with reprex v2.0.2

Note: Next time please provide a proper REPRoducible EXample (reprex) illustrating your issue.