Cannot add an extra column with aggregated data to my original dataset due to number of rows

aggregate summarises the data - it returns only a single row of sum value for each group, we need ave to create a new column

dt1$CAT <- with(dt1, ave(DAMT, RC, FUN = sum))

You could do this also simply using dplyr with a group_by and mutate:

dt1<-structure(list(Population = c("p1", "p1", "p1", "p1", "p1", "p1", 
                                   "p1", "p1", "p1", "p1", "p2", "p2", "p2", "p2", "p2", "p2", "p3", 
                                   "p4", "p5", "p5", "p6", "p6", "p6", "p6", "p6", "p6", "p7", "p7", 
                                   "p7", "p7", "p7", "p7", "p7", "p7", "p7", "p8"), RC = c("p1_1_a", 
                                                                                           "p1_1_b", "p1_2_a", "p1_2_b", "p1_3_a", "p1_3_b", "p1_4_a", "p1_4_b", 
                                                                                           "p1_5_a", "p1_5_b", "p2_a", "p2_b", "p2_c", "p2_d", "p2_e", "p2_f", 
                                                                                           "p3_a", "p4_a", "p5_a", "p5_a", "p6_a", "p6_b", "p6_c", "p6_d", 
                                                                                           "p6_e", "p6_f", "p7_a", "p7_b", "p7_c", "p7_d", "p7_e", "p7_f", 
                                                                                           "p7_g", "p7_h", "p7_i", "p8_1"), DAMT = c(25, 12, 14, 11, 3, 
                                                                                                                                     3, 2, 4, 5, 0, 0, 0, 0, 25, 12, 14, 11, 3, 3, 2, 4, 5, 0, 0, 
                                                                                                                                     0, 0, 25, 12, 14, 11, 3, 3, 2, 4, 5, 0), EU = c(10, 20, 30, 40, 
                                                                                                                                                                                     50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 
                                                                                                                                                                                     190, 200, 210, 220, 230, 240, 250, 260, 270, 280, 290, 300, 310, 
                                                                                                                                                                                     320, 330, 340, 350, 360)), row.names = c(NA, -36L), class = c("tbl_df", 
                                                                                                                                                                                                                                                   "tbl", "data.frame"))


library(dplyr)
dt1 %>% 
  group_by(RC) %>%
  mutate(CAT = sum(DAMT))
#> # A tibble: 36 × 5
#> # Groups:   RC [35]
#>    Population RC      DAMT    EU   CAT
#>    <chr>      <chr>  <dbl> <dbl> <dbl>
#>  1 p1         p1_1_a    25    10    25
#>  2 p1         p1_1_b    12    20    12
#>  3 p1         p1_2_a    14    30    14
#>  4 p1         p1_2_b    11    40    11
#>  5 p1         p1_3_a     3    50     3
#>  6 p1         p1_3_b     3    60     3
#>  7 p1         p1_4_a     2    70     2
#>  8 p1         p1_4_b     4    80     4
#>  9 p1         p1_5_a     5    90     5
#> 10 p1         p1_5_b     0   100     0
#> # … with 26 more rows
#> # ℹ Use `print(n = ...)` to see more rows

^{Created on 2022-07-22 by the reprex package (v2.0.1)}