Let \(\mathbb{H} P^ n(c)\) be an \(n\)-dimensional quaternionic projective space with the maximum \(c\) of the sectional curvatures. It is known that there are two natural twistor spaces \({\mathcal T}_ n\) and \(\mathbb{C} P^{2n +1}\) over \(\mathbb{H} P^ n\). A harmonic map \(\varphi : \Sigma \to\mathbb{H} P^ n(c)\) is called strongly isotropic (resp. a quaternionic mixed pair) if and only if \(\varphi\) can be lifted to a horizontal holomorphic map into \({\mathcal T}_ n\) (resp. \(\mathbb{C} P^{2n + 1}\)), so that \(\varphi\) has the energy \(4\pi d/c\) for some nonnegative integer \(d\). In this paper, the author proves the following theorem: The space of harmonic 2-spheres in \(\mathbb{H} P^ n(c)\) with fixed energy \(4\pi d/c\), which are strongly isotropic or quaternionic mixed pair, is path-connected for all \(n\geq 1\) and \(d \in \mathbb{Z}\). Combining the theorem with the result of \textit{A. Bahy- El-Dien} and \textit{J. C. Wood} [Proc. Lond. Math. Soc., III. Ser. 62, No. 1, 202-224 (1991; Zbl 0679.58019)], one can easily see that the space of harmonic 2-spheres in \(\mathbb{H} P^ n\) with fixed energy \(4\pi d/c\), which are of class (I), (II) or (III), is path-connected for all \(n \geq 1\) and \(d \in \mathbb{Z}\).