df$date <- dmy(df$date)

Filter data for 1 week windows;

filtered_data <- df %>%
  group_by(Hospital_number) %>%
  filter(row_number(date) == 1 | 
  date - lag(date) > days(7)) %>%
  ungroup()

for week 2, change "lag(date) > days(14)

Similarly for month based.

Let me know if this solves the problem.