This is, at present, a theoretical issue (by which I mean SCOTUS has not consistently ruled in such a fashion as to make it unambiguous). One of the common political theories attached to this area is the Unitary Executive Theory. Many legal scholars seem to endorse a unitary executive of some sort, but there is little in the way of consensus on exactly how strong this theory's consequences are and how much power it truly grants the President. Those who tend to advocate for stronger interpretations, with a strictly hierarchical Executive branch with the President at the top, are the ones that normally operate under the label of "(the) Unitary Executive Theory".
The Unitary Executive Theory is a political theory which, in short, holds that since the Constitution expressly invests the executive power in the President (and no one else), then all powers of the executive branch, delegated or otherwise, are held by the President. Some more extreme versions propose that it completely prevents interference by Congress or the Judiciary in the actions of the President, especially on matters of national security. Less extreme beliefs on the consequences include that the Executive branch is immune to suits from itself (e.g. if two agencies have overlapping authority on something, and are in conflict, then they cannot legally resolve this in the courts, as the President, being the singular executive, cannot sue themselves). Somewhere between includes beliefs that independent agencies and counsels are all unconstitutional, as they effectively wield executive power that is severed from the President's exclusive ownership of all such power.
Proponents of a "unitary executive", but not necessarily "The Unitary Executive Theory", have some substantive differences with such imputations. There is reason to believe that the framers of the constitution did not intend a strictly hierarchical Executive. In particular, they envisioned Congress as having the greater powers of any of the branches and to have a certain amount of authority in structuring the executive branch. Congress can create and remove Executive Branch agencies and positions at will, determine the scope and limits of their powers, and even how much they are paid, for example. The "unitary executive" idea can be said to still hold in such frameworks in the sense that the President functions in a role of supervision and guidance over all members of the executive branch.
The distinction between such a "unitary executive" and that proposed by "(the) Unitary Executive Theory" is in how much specific power the president can exercise. It's the difference between "let's not prosecute crime X so much for now" and "drop these specific cases involving X, pursue these ones more vigorously, investigate A because I think he's guilty of Y, and add charges of Z to these other cases".
Myers v. United States (1926) held that the President can set policy decisions and remove any member of the Executive at will (ostensibly for failing to abide by or enact his policies to satisfaction), but subsequent SCOTUS decisions have variably expanded or constrained such power. He presently has the power to unilaterally remove certain members of the Executive branch, but not all. It's unresolved if the President can legally direct that very specific actions be taken by such members of the executive, or potentially anyone else held to be using "executive power". It remains something of a bone of contention of what, if any, authority the President has over independent counsels and agencies, but SCOTUS precedent to date supports their independence and constitutionality, and has largely restricted the President's control to "principle officers" (a somewhat imprecise term mostly designating positions in the Executive not subordinate to anyone except the President and appointed under Advice and Consent from the Senate; anyone supervised by such a person is an "inferior officer").