sigani eobseoseo pyeonjireul mos sseoyo.

없어서 combines the verb 없다 with the connective ending -아서/어서 to express cause or reason.
So 시간이 없어서 means “because I don’t have time.”
If you simply said 시간이 없어요, you’d only be stating “I don’t have time,” without directly linking it to the next clause.

1. Remove the dictionary ending -다 from 없다 to get the stem 없-.
2. According to Korean vowel harmony, since the stem vowel is ㅓ, you attach -어서 (not -아서).
3. Combine them: 없
   • 어서 = 없어서.

The ending -아서/어서 links two clauses by showing that the first clause is the reason or cause for the second.
In 시간이 없어서 편지를 못 써요, it tells us “(I) can’t write a letter” because “(I) don’t have time.”

Korean uses particles like 이/가 to mark the subject of a clause.
Here, 시간이 marks time as the subject of 없다 (“to not exist”).
Although in casual speech you can drop particles, the standard polite form usually includes them for clarity.

• 안 simply negates an action, meaning “do not do.”
• 못 indicates inability, meaning “cannot do.”
  Since the speaker is unable to write due to lack of time, 못 써요 (“cannot write”) is the correct choice.

쓰다 is an ㅡ-irregular verb. When you add a vowel-beginning ending like -어요, you drop the stem’s ㅡ and use the ending’s vowel:
쓰 + 어요 → 써요.

In Korean, if the subject is clear from context (here, the speaker “I”), it is often omitted.
So 시간이 없어서 편지를 못 써요 implicitly means “I don’t have time, so I can’t write a letter.”

The particle 를 marks 편지 as the direct object of the verb 써요 (“to write”).
Without -를, listeners might be unsure which noun is the action’s target.