Try adding an extra line to reset the target of the hyperlink.
Sub AddLine()
ActiveCell.Select
ActiveCell.Copy
ActiveCell.Offset(1, 0).Select
ActiveCell.Select
Selection.PasteSpecial Paste:=xlPasteAll, Operation:=xlNone, SkipBlanks:= _
False, Transpose:=False
Application.CutCopyMode = False
Selection.EntireRow.Hidden = False
ActiveCell.Offset(-1, 0).Select
Selection.ClearContents
ActiveCell.Offset(1, 0).Select
Selection.Hyperlinks(1).SubAddress = "Sheet1!V" & Selection.Row 'New line added
End Sub
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