Based on the level of respect or formality, you can say "Can I have some more side dishes?" in Korean in 3 ways:

• Informal: banchan jom deo julrae? (반찬 좀 더 줄래)
• Formal: banchan jom deo jusilraeyo? (반찬 좀 더 주실래요)
• Very Formal: banchan jom deo jusigetseupnigga (반찬 좀 더 주시겠습니까?)

- can do: hal su ittda (할 수 있다)
- I: nae-ga (내가), je-ga (제가) - omitted here
- some: jom (좀)
- more: deo (더)
- side dish: banchan (반찬)
- to have: gajida (가지다)

Please refer to How to pronounce Korean Alphabet for the English pronunciation in this answer.