Copy File Macro - Prevent from seeing "Chunking Along"

Why are you opening/closing the destination workbook every time you want to copy a row to it?

Open it once, then run through the source data copying what's required and then close it.

Try this.

Sub CopyToBackupFile()
Dim wbDst As Workbook
Dim wsDst As Worksheet
Dim wsSrc As Worksheet
Dim LastRow As Long, i As Long

    Set wsSrc = ActiveSheet

    LastRow = wsSrc.Range("A" & Rows.Count).End(xlUp).Row
    
    Application.ScreenUpdating = False
    
    Set wbDst = Workbooks.Open(Filename:="C:\Users\My\Desktop\Backup.xlsx")
    Set wsDst = wbDst.Worksheets("Sheet1").Select

    For i = 4 To LastRow

        If wsSrc.Cells(i, 1) <> "" Then
            wsSrc.Cells(i, 1).Resize(, 16).Copy wsDst.Cells(Rows.Count, 1).End(xlUp).Offset(1, 0)
        End If

    Next i

    wbDst.Close SaveChanges:=True

    Application.ScreenUpdating = True
    
End Sub