Við látum ekki garðslönguna liggja á grasflötinni yfir nótt.

This is a very common Icelandic pattern:

• láta + object + infinitive

In this sentence, látum garðslönguna liggja means something like:

• we leave the garden hose lying
• we let the garden hose remain lying

So láta here is not just let in the sense of giving permission. It often means leave, allow, or cause something to be in a certain state.

Látum is the present tense, 1st person plural form of láta.

So:

• ég læt = I let / leave
• við látum = we let / leave

The ending -um is the key signal for we.

Usually, no in ordinary Icelandic.

Even though the verb ending already shows person and number, Icelandic normally still uses the subject pronoun in a regular full sentence:

• Við látum ekki ...

If you remove við, it can sound incomplete unless the context makes it very clear, or unless you are using a different construction such as an imperative or a special stylistic ellipsis.

So for a normal statement, keeping við is the safe and natural choice.

In a normal main clause, ekki usually comes after the finite verb.

So the pattern is:

• Við látum ekki ...

This is standard Icelandic word order for negation in a main clause.

A learner can think of it as:

• subject
• finite verb
• ekki
• the rest of the sentence

Because garðslönguna is the direct object of látum, and here the object is in the accusative case.

The dictionary form is:

• garðslanga = a garden hose

But in this sentence we need:

• accusative singular definite = garðslönguna

Compare:

• garðslangan = the garden hose as a subject, nominative
• garðslönguna = the garden hose as an object, accusative

So the sentence uses the form required by the verb.

That is part of the noun’s declension.

Garðslanga is a feminine noun, and in oblique cases its stem changes:

• garðslanga
• garðslöngu-

Then the definite ending is added:

• garðslönguna

This kind of stem change is something you just have to learn with the noun, but it is very common in Icelandic noun inflection.

After láta, Icelandic normally uses the bare infinitive.

So you say:

• láta eitthvað liggja

not

• láta eitthvað að liggja

This is similar to English:

• let it lie
• not let it to lie

So liggja is in the infinitive because it depends on látum, and it appears without að because that is how this construction works.

This is a very important pair to learn.

• liggja = to lie, to be lying, to be located
• leggja = to lay, to put down

So here liggja is correct because the hose is already in a position on the lawn.

Compare:

• Garðslangan liggur á grasflötinni.
  The hose is lying on the lawn.

• Við leggjum garðslönguna á grasflötina.
  We put the hose onto the lawn.

A good shortcut is:

• liggja = state
• leggja = action of putting

Because á can take different cases depending on meaning:

• dative for location
• accusative for movement toward a place

Here the hose is already lying on the lawn, so this is location, not movement. That is why Icelandic uses the dative:

• á grasflötinni = on the lawn

If you were talking about moving it onto the lawn, you would use the accusative:

• á grasflötina = onto the lawn

This location-vs-movement distinction is very important with prepositions in Icelandic.

That ending shows both:

• dative singular
• definite article

So grasflötinni means the lawn in the dative form.

Icelandic usually attaches the to the end of the noun instead of using a separate word. So instead of a separate word for the, you get endings like -inn, -ina, -inni, and so on.

Here:

• grasflöt = lawn
• grasflötinni = on the lawn, in the dative form

Icelandic often uses the definite form when the thing is understood as a specific, known object in the situation.

Here, grasflötinni naturally suggests a particular lawn, such as the lawn at the house. That is very normal.

Using the indefinite form would change the feel of the sentence and could sound less natural unless you really mean a lawn in a more general sense.

Yfir nótt is an idiomatic expression meaning:

• overnight
• for the night
• during the night

It functions almost like an adverbial time expression.

The reason there is no article is that this is a fixed, common phrase. Icelandic often uses time expressions like this without an article when the meaning is general or idiomatic.

So a learner should try to remember yfir nótt as a chunk meaning overnight.

It is most naturally understood as a general habit, policy, or rule:

• We don’t leave the garden hose on the lawn overnight.

Icelandic present tense often works like English present simple in this way. It can describe:

• habits
• routines
• general truths
• standing rules

So this sentence does not have to mean that the speakers are currently dealing with the hose at this exact moment. It can simply express what they normally do or do not do.