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Here is one possible method to include a counter within a string algorithm to eliminate repeated words:

1. Create an empty dictionary to store words as keys and their respective counts as values.
2. Split the input string into individual words.
3. Iterate through each word and check if it exists in the dictionary.
4. If the word does not exist in the dictionary, add it as a key with a count of 1.
5. If the word already exists in the dictionary, increment its count by 1.
6. Only add the word to the output if its count is equal to 1, thus eliminating repeated words.
7. Convert the output back into a string.

Here is an example implementation in Python:

def eliminate_repeated_words(string):
    words = string.split()
    word_count = {}
    result = []

    for word in words:
        if word not in word_count:
            word_count[word] = 1
        else:
            word_count[word] += 1

        if word_count[word] == 1:
            result.append(word)

    return ' '.join(result)

This function takes a string as input and returns a modified string with all repeated words removed. It does this by using a dictionary to keep track of each word's count and only adding the word to the output if its count is 1.