Can vitamin deficiency can affect mood and mental functioning depression lethargy etc? if so, which supplement?

Yes! Vitamin B complex boosts your mood. I also take Vitamin D, fish oil for Omega-3 fatty acids, and St. John's Wort. All four of these things are extremely beneficial. For years I used to be on all kinds of anti-depressants and mood stabilizers that never really worked. I find that the combination of these supplements takes my depression away entirely and in a way that no anti-depressant ever did. (The key is actually remembering to take them every day! I haven't taken them in weeks and my depression came back.)