As far as I know, the sender cannot include a message with the payment. However, if you're looking to attach a payment purpose to a transaction such can be specified by the recipient:
In order to send a payment on the Lightning Network, the recipient must provide an invoice to the sender. The sender then uses the invoice to construct their payment.
The format of the invoice is specified in BOLT #11: Invoice Protocol for Lightning Payments. The invoice may include optional tagged fields. The field tagged with d
may be used to include a short description of the purpose of the payment.