Mi madre echa un puñado de arroz en la olla y luego sigue preparando la merluza.

Both can work, but echar is very common in cooking when you add an ingredient to something.

• echar arroz en la olla = to add/put rice into the pot
• poner arroz en la olla = to put rice in the pot

In this context, echar often sounds a bit more natural because the idea is adding an ingredient, not just placing an object somewhere. It does not have to mean literally throw here.

They sound the same in standard Spanish pronunciation, but they are different words:

• echa = from echar
• hecha = feminine form of the past participle/adjective of hacer

So:

• Mi madre echa arroz = My mother adds rice
• La comida está hecha = The food is done/made

The h is silent in Spanish, so echa and hecha are pronounced alike.

Un puñado de literally means a handful of.

It is a very common way to talk about an approximate amount, especially with food:

• un puñado de arroz
• un puñado de sal
• un puñado de almendras

It usually suggests not an exact measured quantity, just roughly what fits in one hand.

After expressions of quantity or measure, Spanish normally uses de + noun without the article when speaking generally about a substance or material.

So you get:

• un vaso de agua
• un kilo de patatas
• un poco de aceite
• un puñado de arroz

You would only use del if you meant of the rice, referring to some specific rice already identified in context.

With verbs like echar, poner, or meter, Spanish often uses en where English uses in or into.

So:

• echar arroz en la olla
• poner el libro en la mesa
• meter la ropa en la maleta

The idea of movement into the pot is already understood from the verb echar, so en is perfectly natural here.

Not really. Y simply links the two actions, and luego shows the sequence clearly:

• first: she adds the rice
• then: she continues preparing the hake

So y luego means and then or and afterwards. It is very common and natural in Spanish.

Because Spanish uses seguir + gerund to mean to keep doing or to continue doing something.

• sigue preparando = she keeps preparing / she continues preparing
• siguen hablando = they keep talking
• sigo estudiando = I keep studying

So preparando is the correct form here.
This is one of the most common Spanish structures for an ongoing continued action.

It is the gerundio, usually called the gerund in English explanations.

For -ar verbs, the ending is -ando:

• preparar → preparando
• hablar → hablando
• cocinar → cocinando

For -er and -ir verbs, the ending is usually -iendo:

• comer → comiendo
• vivir → viviendo

In this sentence, preparando works with sigue to express a continuing action.

The definite article la is used because it refers to a specific thing: the hake she is preparing.

Spanish often uses the article with foods when talking about a specific item in context:

• preparo la sopa
• cortó el pan
• está haciendo la merluza

Without the article, merluza would sound more general or dictionary-like, not like a specific fish/dish in the situation.

Not necessarily. Preparar is broader than just cook.

It can include things like:

• cleaning the fish
• cutting it
• seasoning it
• getting it ready for cooking
• cooking it as part of the whole process

So sigue preparando la merluza means she continues working on the hake, but the exact stage depends on context.

The present tense in Spanish can be used in a few natural ways here:

• to describe what is happening now
• to describe a habitual action
• to narrate actions vividly, step by step, almost like in a recipe or story

So this sentence could mean:

• what your mother is doing at this moment, or
• what she usually does when cooking this dish

Spanish uses the present tense very often for this kind of description.

Yes. Mi madre is a bit more neutral or standard, while mi mamá sounds more affectionate and everyday in many contexts.

Both are correct:

• Mi madre echa un puñado de arroz...
• Mi mamá echa un puñado de arroz...

In Spain, madre is completely normal, especially in neutral description. Mamá sounds more personal and familiar.