I marked the parts missed or mistranslated below.

- For 미세먼지, 'find dust bearing air' or 'dusty air' might be clearer than 'fine dust' for people not familiar with this term.
 
미세먼지에 이어 내일은 오후 늦게부터 전국에 다소 많은 눈이 내려 쌓일 것으로 보입니다.
Following the fine dust, tomorrow from late afternoon much snow will be seen dropping down across the country.

Following the dusty air, it appears we will have a fairly heavy snowfall and pileup starting tomorrow in late afternoon.

- 다소 많은 눈 = relatively/fairly heavy snow (다소 = relatively; somewhat).

- -으로 보입니다 = it appears; it is forecast/predicted.  (It doesn't mean the snow will be seen falling).

특히 눈이 그친 뒤에는 영하 15도 안팎의 강력 한파가 찾아올 것으로 보여 대비가 필요합니다.
Especially after the snow, strong cold wave about minus 15 degree Celsius is expected to come.

In particular, we advise you to be prepared for a strong cold wave of about 15 below zero which is expected to arrive after the snow.

- 대비가 필요합니다 = preparation is necessary = you are advised to make the necessary preparations.

[기자]
[reporter]

일주일 가까이 이어진 미세먼지는 수도권부터 점차 사라지고 있습니다.
After a week, fine dust in the metropolitan area has gradually disappeared.

After continuing for nearly a week, the dusty air is dissipating gradually starting from the Seoul area.

앞으로 찬 바람이 점차 강해질 것으로 보여 당분간 미세먼지는 쉬어갈 것으로 보입니다.
Before cold wind gradually becomes stronger, fine dust is seen to "rest" for a while.

It appears the dusty air will let up for the time being as cold wind gusts are expected to get stronger in the days to come.

- 앞으로 = going forward; in the days to come; in the future (not "before").

- 당분간 = for the time being; in the foreseeable future.

하지만 이번 주에는 미세먼지 대신 눈과 강력 한파가 바깥 활동을 힘들게 할 전망입니다.
However, in this week instead of fine dust and snow, a strong cold wave is predicted to make outside activity difficult. 
For this week, however, the forecast is that snow and strong cold air will replace the dusty air to make outdoor activities difficult.