What were the relations between the Vijaynagar Kingdom and the Mughal Empire?

While there may have been some exchange between the two states I see one issue here: there was quite little temporal overlap between their actual periods of existence. That's to say that while Vijayanagar rulers nominally stayed in power until the mid 17th century, their influence had been massively reduced following the heavy military defeat against various Muslim states in 1565. On the other hand the Mughals had formed a state by the 1520s, but the state's consolidation is usually dated only to Akbar's rule starting in the 1550s and coming to fruition under him in the later 16th century.

So put another way, while one realm's power had been massively reduced by the mid 16th century (Vijayanagar), the other realm was only starting to really consolidate in this same time frame (Mughals).

Plus looking at geopolitics, Vijayanagar was surrounded by Muslim states leading to adaptation and as mentioned eventual conflict. There was no shared border with the more Northern Mughal empire though, meaning even less likelihood for pronlonged contact and/or conflict between both powers.

Still there may have been smaller scale exchanges after the 1560s until the 1650s that I'm not aware of and hopefully others can speak to. Meantime you may also be interested in my older answer on earlier Muslim influences on Vijayanagar (see also follow up):