yes only gaining the specific immunities to charm, fear and compulsion stops you from being able to cast psychic spells immunity to mind effecting effects does not
Where do the rules say this?
I find it a bit silly for a construct to be able to cast any type of spells (especially Psychic spells, which require emotions). RAW, though, there's no reason a construct couldn't take levels in a Psychic casting class and cast Psychic spells. I actually couldn't even find anything that says a MINDLESS creature can't do this, though there must be something. Right? Obviously, the creature wouldn't be able to be an Int-based caster, but a Wis- or Cha-based caster is totally legal.
Androids can either take the ability to gain emotions or take the Logical Spell feat and pay for metamagic like the NPC in Shattered Star Book 5 who has still spell on all her spells. You can't provide an emotion component if you are emotionless.
Now, whether this applies to constructs? If you can't feel a specific emotion, the rules say:
Emotion components represent a particular emotional state required to cast the spell.
A psychic spellcaster marshals her desire in order to focus and release the spell’s energy. It is impossible to cast a spell with an emotion component while the spellcaster is under the influence of a non-harmless effect with the emotion or fear descriptors. Even if the effect’s emotion matches the necessary emotion to cast the psychic spell, the spellcaster is not in control of her own desires and animal impulses, which is a necessary part of providing an emotion component.
Now are Constructs immune to emotion, fear, or anything that is an emotion? It does say Mind-affecting and Morale. Now, it's not immune to any Emotions besides that (if Morale is counted as an emotion). But we have this FAQ:
What makes something a fear effect? What about a morale effect?
Fear effects include spells with the fear descriptor, anything explicitly called out as a fear effect, anything that causes the shaken, frightened, or panicked condition, and all uses of the Intimidate skill. Intimidate, in particular, is a mind-affecting fear effect, so fearless and mindless creatures are immune to all uses of Intimidate.
Morale effects, unlike fear effects, so far have not had a descriptor or a call-out. Anything that grants a morale bonus is a morale effect. For example, the rage spell grants a morale bonus, so a creature immune to morale effects would be immune to the entire spell, including the –2 penalty to AC.
Nothing in there calling Morale or Fear effects emotion, just Mind-affecting. So, we are at a point where the rules aren't really clear on this subject. By RAW, nothing stops a Construct from casting, unlike an Android, which calls out it isn't affected by emotion effects. Now, should a Construct be able to cast Emotion based spells? That depends on what you want in your game. Or what the GM wants in their game. Talk to your GM/Players about it and find what best works for your group.
And as an aside, I'd allow some Constructs cast them, but not all, as my interpretation is that some Constructs are emotionless husks while others aren't. And I'd likely make this distinction on a case by case basis, where I would take into account the monsters ability scores, lore, and abilities to determine whether it can or not.
Generally, I don't allow mindless anything to also have class levels. Awakened, however, is a fair game.
An android's lack of emotion components for psychic casting figured into the backstory of one of my NPCs. Pathfinder Player Companion: Black Markets describes Tallend Halant, a notable NPC in the Tarnished Halls black market as, "Rogue Technomancer Tallend Halant (CN female android sorcerer 9)." I wwrote a full stat sheet for her. Her feats are Logical Spell (1st), Technologist (3rd), Empathy (5th), Craft Technological Item (7th), Skill Focus (Knowledge[engineering]) (7th bonus), Craft Technological Arms and Armor (9th). I invented her background as:
DESCRIPTION
Tallend Halant is a blue-skinned android with white hair, white lips, and blue nanite markings on her skin. Her demenor is professional and her emotions are subdued, but she is willing to talk at length with strangers about her background or technical matters. Androids age more slowly than humans, and she appears 10 years younger than her current age of 53.
She was born in captivity, a slave to the Technic League. Her previous incarnation had been a nanite bloodline sorcerer in bondage to the Technic League. The Technic League hoped she would manifest those powers again when she rejuvenated. She was taught to be a loyal slave, with the belief that machines like herself were created to serve humans. She developed nanite-bloodline abilities as predicted, but appeared deficient in her spellcasting. Unknown to everyone she cast everything as a Logical Spell because she was born a psychic caster without emotions. The Technic League trained her as support for their field agents.
The cruelties left in the wake of her field team shattered Tallend's loyalties. Pushed to her limits, she manifested emotions, but kept them secret. That boosted her psychic casting and let her escape 30 years ago. She loves freedom almost more than she loves her life.
She found work using her technological training in Hajoth Hakados on the Seven Tears River. Yet even in that android-friendly city she was not safe, for the Technic League offered a bounty to recapture their valuable slave. She teamed up with young Blood Gar smuggler Drakenda Kuldar and moved her business upriver to a secret location. Their trade in refurbished technological items became popular and they set up a black market fair, The Tarnished Halls, in random villages along the Seven Tears River every week. The fair attracts a violent, criminal element. Tallend views them as free to choose violence, but will intervene if it disrupts her commerce.