The noun sense does not.

affect

noun

    Psychology.,  feeling or emotion.

    Psychiatry.,  an expressed or observed emotional response.

    Restricted, flat, or blunted affect may be a symptom of mental illness, especially schizophrenia.

    Obsolete.,  affection; passion; sensation; inclination; inward disposition or feeling.

Now let's replace that in my original phrase:

> prose intending to imitate the affect of poorly composed sentences

becomes

> prose intending to imitate the feeling or emotion of poorly composed sentences

My point was that the author is trying to convey a specific feeling by way of poorly composed sentences. Perhaps they want a colloquial feel or a ranting feel or a rambling one. An obvious example would be the massive run-on sentence in Ulysses.

Poorly constructed sentences cannot have an affect.

Written text can absolutely signify an affect, and poorly constructed sentences regularly do.