DateTime conversion fails when the datetime has seconds
Well, February doesn't have 30 days, so you can't naively just add 1 to the month number and expect to end up with a valid date. (Not to mention leap years, Dec-to-Jan wraparounds that should advance the year, etc.)
I think you'll find it much easier to convert to a date first, then use DATEADD. Then, if you like, convert back to a string format:
DECLARE @origstring VARCHAR(40) = '30/1/2020 20:34:16'
DECLARE @origdate DATETIME, @newdate DATETIME

SET @origdate = CONVERT(DATETIME, @origstring, 103)
SET @newdate = DATEADD(MONTH, 1, @origdate)

SELECT FORMAT(@newdate, N'MM/d/yyyy HH:mm:ss')

Your original string '30/1/2020 20:34:16 6 300' doesn't appear to be a valid date, the problem is the  6 300 at the end. If that's in your source data, just trim that off when you convert:
SET @origdate = CONVERT(DATETIME, LEFT(@origstring, 18), 103)