Filling blank cells based on neighbouring column

I had a quick look at your code... if you're looking to copy the filtered range only perhaps:

Sub Button1_Click()
Dim rngPaste As Range
With ThisWorkbook.Sheets("Sheet1")
    Set rngPaste = .Cells(.Rows.Count,"A").End(xlUp).Offset(1)
End With
Workbooks.Open "C:\Users\u565873\Desktop\TEST source.xlsx"
With ActiveWorkbook.Sheets("Sheet1").Range("$A$7:$D$11")
    .AutoFilter Field:=3, Criteria1:="<>"
    .SpecialCells(xlCellTypeVisible).Offset(1).Copy rngPaste
    .AutoFilter
End With
ActiveWorkbook.Close False
Set rngPaste = Nothing
With ThisWorkbook.Sheets("Sheet1")
    .Range(.Cells(2, "A"), .Cells(.Rows.Count, "A").End(xlUp)).Offset(, 4).SpecialCells(xlCellTypeBlanks).Value = .Range("L1").Value
End With
End Sub