There is nothing in the Constitution that requires CNN, Politico and MSwhatever to be allowed into the Whitehouse.
Here is the constitutional basis that has been adopted by the courts thus far[ this post is not me endorsing these decisions]
“We began by emphasizing that Sherrill’s claim “[wa]s not premised upon the assertion that the White House must open its doors to the press, conduct press conferences, or operate press facilities.” Id. at 129. But given that “the White House has voluntarily decided to establish press facilities for correspondents who need to report therefrom” and given that “[t]hese press facilities are perceived as being open to all bona fide Washington-based journalists,” we held that “the protection afforded newsgathering under the first amendment . . . requires that this access not be denied arbitrarily or for less than compelling reasons.” Id. at 129 (internal citations omitted). Moreover, “the interest of a bona fide Washington correspondent in obtaining a White House press pass” is not only “protected by the first amendment” but also “undoubtedly qualifies as [a] liberty [interest] which may not be denied without due process of law under the fifth amendment.” Id. at 130–131.
Karem v Donald Trump, US Court of Appeals, DC Cir (2020) relying upon Sherrill v. Knight, 569 F.2d 124(D.C. Cir. 1977) [please disregard the informal citations as I am exhausted and am not billing for this ;-) ]