Q:

Are there any psychics available to the Marvel Cinematic Universe? 

During the first season of Agents of SHIELD, the main antagonist was someone calling themselves "The Clairvoyant", who claimed to have psychic powers.  This claim was being constantly questioned by SHIELD agents like Coulson and Fitz/Simmons since, as they repeatedly asserted, "there's no such thing as psychic powers."  

It turns out in this case, they were correct, because "The Clairvoyant" had no actual super powers, merely high-level clearance. 

Clearly, within the Marvel comics universe, there are psychic powers, but the only names I can come up with off the top of my head are all from the X-Men franchise (Xavier, Phoenix, Emma Frost, etc).  For well-known legal reasons, none of those will be appearing in the MCU movies or TV shows any time soon. 
Are there any psychic characters that are available to appear in the MCU, and have any of the existing movies/TV shows name-dropped those characters? 

A:

At this point, there are plenty of psychic characters in the Marvel Cinematic Universe.  
But first, we need to deal with a vexing question.  
What is a “psychic” power? 

Sometimes psychic powers refer only to powers that affect the mind and require mental effort (i.e., they both come from the mind and affect the mind).  By this standard, we have:

Mantis.  She uses powers that can affect the minds of others, such as making them sleep, as well as sensing emotions.  These abilities seem to require effort, as seen when

 she had difficulty restraining Thanos because of his mental strength.  

Stephen Strange and the other masters of the mystic arts.  Among their powers are abilities that affect the mind or perhaps soul, as when

 the Ancient One punched Stephen Strange out of his body. 

Thanos, if he possesses the Mind or Soul Stone.  This is speculative, but either of these stones could affect the mind, and use of the stones seems to require effort.  
Vision.  The Mind Stone seemingly allowed him to 

 access the mind of Ultron, and we saw that it required mental effort on his part.  

Wanda the Scarlet Witch, who is able to read and control minds, as well as creating mental illusions, with her powers, which again seem to require mental exertion on her part.  

If we relax the constraint that the powers must come from the mind, we also have:

Kilgrave, who was able to control minds through a virus. 
Lorelei, who could control the minds of male Asgardians and humans (or only female-attracted individuals thereof?)  through her voice and contact.  
Ben, an Inhuman from the future who heard thoughts, but whose mechanism was not explained.  It’s not clear whether he needed to exert mental effort to do so.  
Hive, who could control the minds of Inhumans.  
Loki, whose illusions often seem to affect the minds of his targets, rather than manipulating light.  Or Loki while using the Mind Stone to control  people.  

If we consider powers that are frequently called psychic, such as precognition and telekinesis, we can add:

Reina, an Inhuman who was able to see the future in dreams.  
Robin, an Inhuman who saw the future all the time. 
And Charles, her father, who could cause others to see the future when he touched them. 
Ebony Maw, a sorcerer under Thanos’s command who could employ telekinesis. 
Giyera, an Inhuman with the power of telekinesis.  

And many others, depending on what one considers a stereotypically “psychic” power.  
If we take psychic powers to be any powers called psychic, very few of the previous qualify.  Mantis, for example, is usually described as an “empath,” and Robin is usually called a “seer.” 

These lists are not exhaustive: there are likely many other characters in the MCU who would qualify as psychic by these definitions.  
As always, the term “psychic” is rather vague.  Like most terms in the Marvel Cinematic Universe (“science,” “quantum,” “alien”), it seems to be used as a substitute for what we would simply call “magic.”  If we take psychic powers to be any powers that come from the mind (the broadest definition), virtually any power in the MCU that isn’t directly beating people up, technological, or a passive defensive ability counts, such as Thor’s lightning, Daisy’s earthquake powers, etc. 

A:

In Captain America 2, Agent Sitwell specifically namedrops Dr Stephen Strange. 
One could argue that he is not actually a psychic, per se - but that he has psychic abilities via his control of the mystic forces. 
Marvel's own page lists his powers:

Doctor Strange is one of the most powerful sorcerers in existence.  Like most sorcerers, he draws his power from three primary sources: the invocation of powerful mystic entities or objects, the manipulation of the universe's ambient magical energy, and his own psychic resources.  Strange's magical repertoire includes energy projection and manipulation, matter transformation, animation of inanimate objects, teleportation, illusion-casting, mesmerism, thought projection, astral projection, dimensional travel, time travel and mental possession, to name a few.  The full range of his abilities is unknown.  Doctor Strange's powers are sometimes less effective against strictly science-based opponents, although he can overcome this limitation with effort. 

ComicVine has a great writeup about him and explains further:

Dr. Strange uses mystical energy invoked upon by spells/incantations or by the power of divine beings to manipulate forces of the universe including projection of mystic energy as bolts, slashes, binds, shields, or a scattered array of energy appearing like electricity, manipulating matter and various energies (physical or spiritual) even transmutation of matter, controlling the elements, opening or sealing mystical portals to other realms, teleportation or dimensional travel, reality warping and control over dimension, seeing through or casting illusions, invisibility and intangibility through astral projection, vast mental powers such as telepathy and telekinesis, communication with the dead and spiritual entities, seeing the future or past, manipulating dreams, removing spirits or placing them back into bodies, summoning supernatural beings, or using the powers of supernatural items. 

The last highlighted portion fits quite well with the typical definition of a "clairvoyant", particularly with the powers that The Clarivoyant reportedly had. 

A:

There are not only psychic characters available, they could easily have been included in a forthcoming Marvel movie.  Vance Astro and Mantis of the Guardians of the Galaxy are both psychic, but don't appear to actually be in the Guardians of the Galaxy movie.