¡Su niño es un mocoso malcriado! Le da todos los caramelos que quiere en el mercado, y cuando
él se agarra un berrinche, ella se da por vencida.

That's how I would say it in gringoese. (g)

Gustavo, I understand your Spanish, but forgive me, I do not understand your English. What is "bogger boy"''?

I think "niño malcriado" or "niño consentido" is what I was looking for -- gracias!

¡Su hijo/crío,/niño es un mocoso malcriado! Le da todos los caramelos que quiere en la tienda, y cuando él coge un berrinche, ella cede.

Ella se da por vencida is not the same as give in, in my opinion. It literally means: to accept defeat. But in this case it means she will do again what the child wants, so she "cede".

The other changes are simply Spanish preferences (from Spain).

Muchas gracias a todos. ¡Espero que mi bebé tan bonito no se crezca a ser un mocoso malcriado! Pues tiene menos de seis meses, todavía está monono, ¿no?

I actually used cede at first, but decided that the register didn't match that of "give in," so I changed it. I know that darse por vencido means give up, rather than give in, but the meanings are very close, and I thought the register was closer that way. That is, I was thinking that "concede" is more formal in register than "give in." I guess that is not the case with the Spanish equivalents.

Thanks for the comments.