If you want a specific error handling, then this is what I recommend:

Sub RefreshConnectionsWithErrorHandler()
    Dim cn As WorkbookConnection
    Dim isPowerQueryConnection As Boolean
    Dim errMsg As String

    On Error GoTo ErrorHandler  ' Enable error handling for the entire sub

    ' Loop through all connections
    For Each cn In ActiveWorkbook.Connections
        isPowerQueryConnection = InStr(1, cn.OLEDBConnection.Connection, "Provider=Microsoft.Mashup.OleDb.1") > 0
        
        ' Refresh each connection
        If isPowerQueryConnection Then
            cn.OLEDBConnection.BackgroundQuery = False ' Disable background refresh for better error visibility
            
            On Error Resume Next ' temporarily ignore errors within the single connection refresh
            cn.Refresh
             On Error GoTo ErrorHandler ' re-enable normal error handling
            
            ' Check for an error condition by checking if the connection was successfully refreshed.
             If Err.Number <> 0 Then
                errMsg = "Error refreshing connection '" & cn.Name & "': " & Err.Description
                Debug.Print errMsg
            Else
                Debug.Print "Connection '" & cn.Name & "' refreshed successfully."
            End If

            Err.Clear ' Clear the error, so you do not get the same error for multiple connections.
        Else
             Debug.Print "Skipping non Power Query connection: " & cn.Name
        End If
    Next cn

    MsgBox "All connections processed. Check the Immediate Window for details.", vbInformation
    Exit Sub

ErrorHandler:
    ' Handle general errors
    MsgBox "An unexpected error occurred during refresh. Error: " & Err.Description, vbCritical
End Sub

• The code iterates through each connection in the active workbook using For Each cn In ActiveWorkbook.Connections.

• It checks for a Power Query connection using a substring search within the connection string.

• cn.OLEDBConnection.BackgroundQuery = False is set before refreshing for better error handling.

• The ErrorHandler catches any unexpected errors and displays a message box.

Error Checking and Handling:

• The Err.Number is evaluated after each attempted refresh.

• If Err.Number is not 0, an error message and the connection name are printed to the Immediate Window.

• If no error occurs, a success message is printed to the Immediate Window.

• The Err.Clear statement clears the error, preventing the same error from being reported multiple times.