Learn Finnish: Lapsi peseytyy itse. - The child washes himself.

It is derived from the basic transitive verb pestä (to wash). By adding the reflexive suffix -ytyä to the verb's stem (pese-), it becomes peseytyä, meaning "to wash oneself". The ending -yy is just the standard third-person singular present tense marker.

If peseytyy already means "washes himself", isn't the word itse redundant?

Grammatically, the verb peseytyy is enough to show the action is reflexive. However, adding itse (self) is very common to add emphasis. In this sentence, it emphasizes independence: the child washes by himself or without help, rather than just indicating who is getting clean.

Why is the suffix -ytyä here instead of just -tyä?

When attaching the -tua/-tyä suffix to a verb stem that ends in -e (like pese-), Finnish naturally adds a -u- or -y- before the suffix to make the word easier to pronounce. Because pestä is a front-vowel word, it takes -y-, resulting in peseytyä rather than pesetyä.

Could I use this verb if the child was washing a dog instead of himself?

No. Reflexive verbs formed with -tua/-tyä are intransitive, meaning they cannot take a direct object. If the child is washing something else, you must go back to the basic verb pestä, for example: Lapsi pesee koiran (The child washes the dog).