Даже если мне некогда, я хотя бы поглажу блузку и проверю, нет ли складок.

Даже если means even if.

• если = if
• даже если = even if, even in the case that

So даже adds a stronger concessive feeling: the speaker means that this will happen even under an inconvenient condition.

Compare:

• Если мне некогда, я не пойду. = If I don’t have time, I won’t go.
• Даже если мне некогда, я хотя бы... = Even if I don’t have time, I’ll at least...

Because this is an impersonal construction.

Russian often expresses states and situations with:

• a dative experiencer +
• a predicative word

So:

• мне холодно = I’m cold
• мне трудно = It’s hard for me
• мне некогда = I have no time / I’m too busy

Literally, мне некогда is something like to me, there is no time.

That is why you get мне, not я.

Here некогда is a fixed word meaning there is no time / no time to do something.

So:

• мне некогда = I don’t have time

This is different from не когда as two words, which would only happen in a contrast like:

• Не когда, а где? = Not when, but where?

So in your sentence, it must be некогда as one word.

Here хотя бы means at least.

It shows the minimum the speaker is willing or able to do:

• я хотя бы поглажу блузку и проверю...
  = I’ll at least iron the blouse and check...

So the idea is:

Even if I’m short on time, I’ll still do this minimum amount.

Note that хотя бы does not mean although here. It can have different uses in different contexts, but here it clearly means at least.

Поглажу is the 1st person singular future of the perfective verb погладить.

That matters because Russian aspect changes how the action is presented:

• гладить = imperfective, process/habit
• погладить = perfective, one completed action

So:

• глажу = I am ironing / I iron
• буду гладить = I will be ironing
• поглажу = I will iron it (as a completed action)

In this sentence, the speaker is talking about a single future action they intend to complete, so the perfective future поглажу is natural.

Also, the prefix по- often suggests a limited, one-time action, which fits the idea of doing at least a quick pass.

Yes, погладить can mean both, depending on the object.

For example:

• погладить кошку = to pet/stroke the cat
• погладить рубашку / блузку = to iron the shirt / blouse

So with clothing, погладить means to iron.

The object tells you which meaning is intended.

Because блузку is the accusative singular, and it is the direct object of поглажу.

The noun is:

• nominative: блузка
• accusative singular: блузку

This is the normal pattern for many feminine nouns ending in -а:

• книга → книгу
• рубашка → рубашку
• блузка → блузку

So поглажу блузку means I’ll iron the blouse.

Because it is another action the speaker plans to do in the future.

• проверю = I will check
• it is the 1st person singular future of the perfective verb проверить

So the structure is:

• я поглажу... и проверю...
  = I’ll iron... and check...

Both verbs are perfective future because both actions are presented as single completed steps.

This is a very common Russian pattern.

нет ли introduces an indirect yes/no question:

• проверю, нет ли складок
  = I’ll check whether there are any wrinkles/folds

Literally, it is closer to:

• I’ll check whether there aren’t any wrinkles

But in natural English, that usually becomes:

• I’ll check if there are any wrinkles
• I’ll check whether there are any wrinkles
• I’ll check that there aren’t any wrinkles

A few key points:

• нет = there is/are not
• ли marks an indirect yes/no question
• ли usually comes right after the word it attaches to, so: нет ли

Because the speaker is checking for the absence of something undesirable.

Wrinkles/folds are unwanted, so Russian naturally says:

• проверю, нет ли складок
  = I’ll check whether there are no wrinkles / whether there aren’t any wrinkles

This is more idiomatic than a neutral existence question.

You could say:

• проверю, есть ли складки

but that sounds more like a neutral check for whether wrinkles exist at all.

So:

• нет ли складок = more natural when you hope there are none
• есть ли складки = more neutral, less idiomatic here

Because нет requires the genitive.

The basic plural is:

• nominative plural: складки

But after нет, Russian uses genitive:

• нет складок

So in the sentence:

• нет ли складок

you get складок, which is the genitive plural.

This is a standard rule:

• есть ошибки = there are mistakes
• нет ошибок = there are no mistakes

There are two commas for two different reasons.

1. Даже если мне некогда, ...
   The first comma separates the subordinate clause introduced by даже если.

2. я хотя бы поглажу блузку и проверю, нет ли складок
   The second comma introduces the indirect question нет ли складок, which depends on проверю.

So the structure is:

• [Even if I don’t have time], [I’ll at least iron the blouse and check], [whether there are any wrinkles].

Also, there is no comma before и because поглажу and проверю share the same subject я.

In this sentence, хотя бы is most naturally understood as applying to the whole minimum plan that follows:

• я хотя бы поглажу блузку и проверю, нет ли складок

That means something like:

At minimum, I’ll iron the blouse and check for wrinkles.

So it is not usually understood as:

• I’ll at least iron it, and then separately I’ll check...

Instead, it frames the following actions together as the minimum effort the speaker will make.