  The Thue--Morse sequence is a prototypical automatic sequence found in
diverse areas of mathematics, and in computer science. We study occurrences of
factors $w$ within this sequence, more precisely, the sequence of gaps between
consecutive occurrences. This gap sequence is morphic; we prove that it is not
automatic as soon as the length of $w$ is at least two, thereby answering a
question by J.~Shallit in the affirmative. We give an explicit method to
compute the \emph{discrepancy} of the number of occurrences of the block
$\mathtt{01}$ in the Thue--Morse sequence. We prove that the sequence of
discrepancies is the sequence of output sums of a certain base-$2$ transducer.
