Although the daily mean temperature in Nagasaki in June is only 73.4°F (23 °C), precipitation is higher in June than in any other month of the year, so you're unlikely to avoid rain. In Kagoshima, the daily mean temperature is 74.5°F (23.6 °C), and precipitation is highest in June, significantly higher than in Nagasaki. That said, it sounds like you'll only have one or two days to spend in Nagasaki, so you could get lucky and miss the rain. If your entire trip is only 7 days long, your plan to spend 4 nights in Fukuoka, taking day trips to both Kagoshima and Nagasaki, may be overly ambitious, but not impossible. It's not clear to me whether your four nights in Fukuoka City include day trips to Kagoshima and Fukuoka, or if those day trips come after a 4-night stay there. If the latter, you'll probably only have one full day in each of those cities. Since it is June, I wouldn't be overly concerned about the heat, though keeping your clothes dry from humidity and rain while traveling, as mentioned in other posts above, is worth considering.