May tanghalian pa sa lamesa, pero kakainin lang namin ito mamaya.

May is an existential marker meaning “there is/are.” In this sentence it introduces the existence of lunch on the table. It’s not a verb meaning “to have,” though English often translates it that way.

• Correct: May tanghalian pa sa lamesa. = “There’s still lunch on the table.”
• You don’t use ang after may: ✗ May ang tanghalian… is ungrammatical.
• You’ll also hear mayroon/meron (variants of the same word). Rough guide:
  • Use may before straightforward nouns: may tanghalian, may pagkain.
  • Use mayroon/meron before pronouns or when you want to pause/emphasize: Meron pa ito sa lamesa, Meron pa kami.

Pa is an enclitic meaning “still/yet/remaining” or “in addition.” Here it means “still.”

• Placement: enclitics like pa normally go in second position in the clause (after the first word or prosodic unit). That’s why it comes after tanghalian here.
• Acceptable: May tanghalian pa sa lamesa.
• Not used as: ✗ May pa tanghalian…
• With emphasis/contrast you might hear pa rin (“still (as before)”): May tanghalian pa rin sa lamesa.