As it says that they (Karna and Wilhelm) where not married at the time of the birth. Wilhelm is only "udlagte barnefader" which means that he is probably only mentioned as the father because the mother said so, there did not need to be any "proof" of this.
If the man "accepted" to be the father, then he would need to pay money to the child. Then there would be records in the local archives for Ullerup for this arrangement, which might give you more info about Wilhelm.
Just an idea.http://danishgenealogy.homestead.com/danishgenealogy.html