Learn Russian: У меня нет настроения читать.

In Russian, you don't usually use a direct verb for "to have." Instead, you say "by me there is no..." using the preposition У (by/at) plus a pronoun in the genitive case. меня is the genitive form of Я (I). Together with нет, У меня нет is the standard way to say "I don't have."

Why is the word for "mood" spelled настроения instead of настроение?

The word нет (there is no) is a negative word that always forces the noun following it into the genitive case to show what is missing. The dictionary form is the neuter noun настроение. To make it genitive, the final -е changes to -я, resulting in настроения.

Can I just attach the infinitive verb читать to the end of the sentence like that?

Yes! Just like in English where you say "mood to read," Russian allows you to use an infinitive verb (a dictionary-form verb ending in -ть, like читать) directly after a noun phrase to describe the action associated with it.

How would I change this sentence to say "He has no mood" or "She has no mood" using the pronouns from this lesson?