Conque
Conque is a conjunction that is used to introduce consecutive sentences, those which introduce a consequence of a fact.Ya he limpiado el baño conque ahora te toca a ti limpiar la cocina.
I already cleaned the bathroom so now it's your turn to clean the kitchen.
Han jugado al fútbol toda la tarde conque, ahora, hagan la tarea.
You have played soccer all the afternoon so, now, do your homework.
Other conjunctions that can replace conque in consecutive sentences are luego, así que, and por consiguiente.
Con que
Con que is a sequence of two words:- either the preposition con and the relative pronoun que. It is normaly used with the determinate articles el, la, los, las that refer to the direct object of the main sentence, but they are optional.
El arma con (la) que mataron a aquella persona se etiquetó como prueba para el juicio.
The weapon with which they killed that person was tagged as evidence for the trial.
- or the preposition con and the subordinate conjunction que. It's different from conque because there's no consecutive value and usually the preposition por is required by a precedent verb, noun, or adjective.
Es suficiente con que me ayudes por una hora.
It's enough to help me for an hour.
Con qué
Con qué is a sequence of two words, the preposition con and the interrogative/exclamative pronoun qué. It is used always in questions (direct or indirect) and exclamations.¿Con qué salsa cocinaste la pasta el otro día? ¡Estaba buenísima!
With what sauce did you cook pasta the other day? It was delicious!
¡Con qué pintas vienes a trabajar! ¡Eso no se lleva a una oficina!
Why are you dressed like that! You don't wear that in an office!
No sé con qué producto has limpiado el suelo pero sigue sucio.
I don't know with what product you cleaned the floor but it's still dirty.