Understanding the basic relationship between one note and another (intervals) is important as you learn to read and play music. Certain notes blend better with others: ‘pleasant' sounds versus ‘unpleasant' sounds. A melody with no ‘unpleasant' sounds would be quite boring after a while.
In your pieces, you can think about harmony as supporting your melody. But remember, your left hand harmony offers its own secondary melody that we want to hear, even in its supporting role that provides texture and interest.
Music is the art of sound moving over time. Harmony is an equal piece of the puzzle to both melody and rhythm. Every note of a melody depends completely on its harmonic relationships with another. Additionally, rhythm is not secondary: it creates an essential structure that ties music together. Without rhythm, a collection of sounds do not result in music. Plus, a melody is completely dependent on (an outcome of) harmonic information.