Husserl insists on two "reductions" in his pure phenomenology. The second reduction is a separation of the existence of ourselves and our attitudes and "their observable essences (Taylor Carman , Forward for Being and Time). Is Husserl's definition of essences the same as Heidegger's view of "being/the essence of being"?
Absolutely not. Heidegger's "essence of Dasein" is really a misnomer to make a point, by stating that Dasein's essence is existence he upends the traditional use of "essence" as form, idea, the opposite of existence. Heidegger questioned that essences can be the kinds of universal invariants that Husserl wanted them to be, and therefore that his "eidetic reduction" that supposedly extracts them by discarding existential and interpretive backdrops can ever succeed.
Husserl, on the other hand, uses essences (a.k.a. ideas, eidoses, forms, etc.) in a much more traditional sense that goes back to Plato's, or perhaps more Aristotle's, forms, invariances of becoming. Except that unlike Aristotle and Plato Husserl is not a realist, he suspends judgements about existence, so the forms are viewed not as pre-existing out there, but as constituted in intentional acts, in which we are directed at them. A specific feature of Husserlian conception is that not only universals like "truth" and "material thing" have essences, but also what Aristotle called abstract particulars, like "this circle" and, this is Husserl's innovation, even empirical particulars like "this house" and "this tree" have them.
Husserl's argument for extending the notion of essence this way was that the structure of the intentional acts in which essences are constituted is uniform, so if they feature in reflective acts of abstraction/generalization there has to be an analog of them in perceptive acts as well. Otherwise, we would not be able to recall or think of the same concrete tree at different times. Closely related to essences is also specifically Husserlian notion of "noema", the "content" of an essence as it appears in an intentional act upon the phenomenological reduction (suspension of beliefs).
Generally, Husserl is much better at describing what the essences are not than what exactly they are. They are not spatio-temporal, yet they accompany and reveal themselves through perceptual objects that manifest in space and time, they are not sense data or mental constructs, but neither are they Plato's reified idealities. See more in Zhok's Ontological Status of Essences in Husserl's Thought, a more elementary source is Follesdal's short Introduction to Phenomenology.