Aggiungo aglio alla zuppa di lenticchie.

In Italian, aglio (garlic) is an uncountable noun when used in a general sense, so you omit the article. Saying Aggiungo aglio simply means “I add garlic.”
You can use the partitive to emphasize “some garlic”:
• Aggiungo dell’aglio = “I add some garlic.”

When you combine the preposition a (to) with the definite article la (the) before a feminine singular noun, you fuse them into alla.
So alla zuppa means “to the soup,” because you’re adding garlic to a specific soup.

• Zuppa di lenticchie (“lentil soup”) uses di to indicate the main ingredient (the soup is made of lentils).
• Zuppa con lenticchie means “soup with lentils,” emphasizing lentils as an added component rather than the defining feature.
In Italian dish names, di is the standard preposition to signal the primary ingredient.

You’re talking about many individual lentils in the soup. In Italian, foods made from small, countable items usually use the plural form:
• lenticchie (lentils)
Using lenticchia (a single lentil) would sound odd.

Aggiungere is a second-conjugation verb (-ere). Its present-tense forms are:
• io aggiungo
• tu aggiungi
• lui/lei aggiunge
• noi aggiungiamo
• voi aggiungete
• loro aggiungono

The gg is a geminated (doubled) consonant, held slightly longer than a single g. It’s a hard g (as in “go”), sustained briefly:
• g in gatto vs. gg in aggiungo.