charAt and substring

import java.util.ArrayList;
import java.util.Arrays;
import java.util.HashSet;

class PigLatinTranslator {

    private static final HashSet < Character > vowels = new HashSet < Character > (Arrays.asList('a', 'e', 'i', 'o', 'u'));
    private static final HashSet < String > specials = new HashSet < String > (Arrays.asList("xr", "yt"));

    public String translate(String phrase) {
        ArrayList < String > piggyfied = new ArrayList < String > ();
        var words = phrase.split(" ");

        for (String word: words) {
            if (vowels.contains(word.charAt(0)) || specials.contains(word.substring(0, 2))) {
                piggyfied.add(word + "ay");
                continue;
            }

            var length = word.length();
            
            for (int pos = 1; pos < length; pos++) {
                var letter = word.charAt(pos);
                if (vowels.contains(letter) || letter == 'y') {
                    if (letter == 'u' && word.charAt(pos - 1) == 'q') pos += 1;
                    piggyfied.add(word.substring(pos) + word.substring(0, pos) + "ay");
                    break;
                }
            }
        }
        return String.join(" ", piggyfied);
    }
}

This approach starts by importing from packages for what will be needed.

The class defines two private static final HashSets to define the vowels and special letter sequences. They are private because they don't need to be directly accessed from outside the class, and final because they don't need to be reassigned after being created. They are static because they don't need to differ between object instantiations, so they can belong to the class itself.

The translate() method starts by defining the ArrayList() to hold the words to be output. The input String is then split() on a space to create the array of words.

A for-each loop is used to iterate the words.

An if statement checks if the first character in the word is a vowel. It does this by using the String charAt() method to get the character at the first index of the word. It then uses the HashSet contains() method to see if the HashSet of vowels contains the character. If so, the logical OR operator (||) short circuits and ay is concatenated to the word, which is then added to the output ArrayList, and the loop continues.

If the first character is not a vowel, the other side of the OR operator checks if the first two characters of the word are a special sequence. It does this by using the String substring() method to get a String of the first two characters of the word. It then uses the HashSet.contains() method to see if the HashSet of special sequences contains the substring. If so, then ay is concatenated to the word, which is then added to the ArrayList, and the loop continues.

If the word does not start with a vowel or special sequence of letters, then the length of the word is set to prepare for iterating through the word characters, starting with the second character at index 1.

Inside a for loop, the character at the current iterating index is set. The HashSet.contains() method is used to see if the HashSet of vowels contains the character. If not, the other side of the logical OR operator checks to see if the character is a y. If the character is a vowel or y, then the character is checked to see if it is a u, and, if so, if the previous character is a q. If the index is at the end of a qu sequence, then the index is incremented by 1, otherwise the index is left as is. Finally, the word is made into Pig Latin by getting a substring() from the index until the end of the word, to which is concatenated a substring() from the beginning of the word up to but not including the indexed character, and concatenating ay.

After all of the words are iterated, the ArrayList of words is assembled into a single String by using the String.join() method to join all of the words together, separated by a single space, and that String is returned from the function.