I think you might be confusing shift in supply and increase in supply of apples in your question.
You are completely right that shift in demand of apples will cause more apples to be supplied to the market. As we can see from the picture below in response to shift in demand, price increases and quantity supplied also increases.
However, supply curve does not shift. We are moving along the supply curve but supply curve itself is fixed in this case. Supply curve represents the firm supply at any arbitrary price. Why would discovery of apples health benefit make suppliers to supply more at any price? Suppliers are not charities, they won’t increase their output at any price just because that would benefit consumers.
Also, market competition does not necessarily increase proportionally with number of firms. If there are 10000 firms and number will change to 10010 firms the level of competition likely remains the same. It is also not guaranteed that the number of firms will change as the shift in demand and increase in quantity might be accommodated by existing firms. Furthermore, market for apples should already be operating under approximately perfect competition since it’s homogenous product with many sellers, thus increase in number of sellers should not be expected to change level of competition at all.
While you could probably construct some convoluted example where competition would be affected, it would represent some special case that’s not representative of what would happen in most cases described by such question.
You should not overthink these simple 101 high-school economics questions. Even if there might be some rare counter examples you should interpret the question as asking what would typically happen to supply conditional on information given.