Here is some code that takes a column that is of the type character and has the form yy-mm-dd, turns it into a date and then makes a new column of characters that have the form mm-dd. I included calls to str() to show how the process is working. It is not possible, as far as I know, to have a Date consisting of only the month and the day.
What are you trying to accomplish with the mm-dd column?
DF <- data.frame(Date = c("20-04-03", "20-05-01", "20-06-13"))
str(DF)
#> 'data.frame': 3 obs. of 1 variable:
#> $ Date: chr "20-04-03" "20-05-01" "20-06-13"
library(lubridate)
#>
#> Attaching package: 'lubridate'
#> The following objects are masked from 'package:base':
#>
#> date, intersect, setdiff, union
DF$Date <- ymd(DF$Date)
str(DF)
#> 'data.frame': 3 obs. of 1 variable:
#> $ Date: Date, format: "2020-04-03" "2020-05-01" ...
DF$NewDate <- format(DF$Date, "%m-%d")
str(DF)
#> 'data.frame': 3 obs. of 2 variables:
#> $ Date : Date, format: "2020-04-03" "2020-05-01" ...
#> $ NewDate: chr "04-03" "05-01" "06-13"
Created on 2020-08-26 by the reprex package (v0.3.0)