For a suitable triangulated category [Formula: see text] with a Serre functor [Formula: see text] and a full precovering subcategory [Formula: see text] closed under summands and extensions, an indecomposable object [Formula: see text] in [Formula: see text] is called Ext-projective if Ext[Formula: see text]. Then there is no Auslander–Reiten triangle in [Formula: see text] with end term [Formula: see text]. In this paper, we show that if, for such an object [Formula: see text], there is a minimal right almost split morphism [Formula: see text] in [Formula: see text], then [Formula: see text] appears in something very similar to an Auslander–Reiten triangle in [Formula: see text]: an essentially unique triangle in [Formula: see text] of the form [Formula: see text] where [Formula: see text] is an indecomposable not in [Formula: see text] and [Formula: see text] is a [Formula: see text]-envelope of [Formula: see text]. Moreover, under some extra assumptions, we show that removing [Formula: see text] from [Formula: see text] and replacing it with [Formula: see text] produces a new subcategory of [Formula: see text] closed under extensions. We prove that this process coincides with the classic mutation of [Formula: see text] with respect to the rigid subcategory of [Formula: see text] generated by all the indecomposable Ext-projectives in [Formula: see text] apart from [Formula: see text]. When [Formula: see text] is the cluster category of Dynkin type [Formula: see text] and [Formula: see text] has the above properties, we give a full description of the triangles in [Formula: see text] of the form [Formula: see text] and show under which circumstances replacing [Formula: see text] by [Formula: see text] gives a new extension closed subcategory.