This is the updated version Below:
Sub Update_Sheets()
Dim sh1 As Worksheet, sh2 As Worksheet
Dim i As Long, j As Long, f As Range
Set sh1 = Sheets("NON_PMI")
For i = 2 To sh1.Range("A:A").Find("*", , xlValues, , xlByRows, xlPrevious).Row
Set sh2 = Sheets(sh1.Range("A" & i).Value)
Set f = sh2.Range("B:B").Find(sh1.Range("B" & i).Value, , xlFormulas, xlWhole)
If f Is Nothing Then
j = sh2.Range("B:B").Find("*", , xlValues, , xlByRows, xlPrevious).Row + 1
sh2.Range("B" & j).Value = sh1.Range("B" & i).Value
sh2.Range("C" & j).Value = sh1.Range("C" & i).Value
End If
Next
End Sub