ArrayIndexOutOfBoundsException when finding words in an array that and with a specific letter

You've got too much code for the task, which has lead to a bug creeping in and hiding in plain sight. As a general rule, keeping code as simple as possible results in less bugs.

Delete this, which you don't need.

for (int i = 0; i < words.length; i++) {
    words[i] = words[i] + " ";
}

And delete all of this:

for (int j = 0; j <= words[i].charAt(j); j++) {
    if( words[i].charAt(j) == 'a' && words[i].charAt(j + 1) == ' '){
        System.out.println(words[i]);
    }
}

instead basing your code on endsWith("a"):

for (String word : words) {
    if (word.endsWith("a")) {
        System.out.println(word);
    }
}

which is easy to read and understand (and thus easier to avoid bugs).

Even simpler, since you don't need to reference the array, use the result of the split directly:

String text = sc.nextLine();

for (String word : text.split(" ")) {
    if (word.endsWith("a")) {
        System.out.println(word);
    }
}