How to convert monthly data to weekly data in SQL (SSMS)?

I have a table in which I have monthly data for example:

If the 1st of the month falls on the weekend, then the first Monday of the month is considered as date. This is the reason for which in April I have the 3rd instead of the 1st.

I need to convert this monthly data to a weekly data in the following way: for each Monday in each month a new line will be inserted such as:

I know I should use some type of recursive function to obtain the result but I am not sure how. Any help is much appreciated.

What you can do is generate a small calendar-like table inside a Common Table Expression that lists out your dates and enough information to select just the ones you want, then select from that CTE.
In my SQL Fiddle example http://sqlfiddle.com/#!18/2ff909/12 I use the sys.columns table to generate a small Tally table (just 31 sequential integers) which I add to the dates. I calculate their month and day of week. Then I select from the CTE based on your Monday criteria, and making sure the new date is for the same month. If you were going to do more than 1 year at a time, you would need to include the year value as well.

insert Dates (theDate, theValue)
values ('20230201',1)
     , ('20230301',2)
     , ('20230403',3);
with Tally as (
   select top(31) row_number() over (order by column_id) as N
     from sys.columns
), ExpandedDates as (
   select Dates.theDate
        , Dates.theValue
        , datepart(month,Dates.theDate) as dateMonth
        , dateadd(day,tally.N-1,Dates.theDate) as newDate
        , datepart(month,dateadd(day,tally.N-1,Dates.theDate)) as newMonth
        , datepart(weekday,dateadd(day,tally.N-1,Dates.theDate)) as dayOfWeek
     from Dates
   cross join Tally
)
select theDate
     , newDate as Monday
     , theValue
  from ExpandedDates
 where dayOfWeek = 2
   and dateMonth =  newMonth
order by theDate, newDate