The amount of throwing/practicing in your off-season all depends on what your goals are. Those who just play rec ball and are content with it, are likely not to throw at all during the off-season. For those who play at a higher level, or wish to significantly improve their game, do practice plenty in the off-season. How much greatly depends on one's goals and understanding of how hard work pays off.
I live up north as well (Cincinnati area). My son's season is typically April-July. Fall Ball from mid-August thru mid-October. One month no throwing, then off-season training starting late Nov. leading up to Spring. My son's off-season training involves all aspects of the game, not just pitching. He plays for a high level team, so off-season training is an absolute necessity if he wants to stay ahead of the pack. Taking 4 months off is out of the question.
IMO, at 9/10 years of age, throwing twice a week during this time of year is good. You're doing more than most. My son just turned 14 and is throwing 3x a week. To see significant improvement, I would continue to throw at least 2x a week and set goals. Examples of some pitching goals for this age would be
1. Improved mechanics (really any age for this). Perhaps post a video on this site for recommended changes. Incorporate drill work into your practices.
2. Improved command. Being able to just throw strikes at 9 can sometimes be a challenge. If your son can learn to not only throw strikes but learn to locate them, he'll have a strong advantage. During some of your bullpens, give your son a specific target (e.g. low, outside corner) for the entire time. Count how many times he hits it out of 20 or 40, or whatever. Log it and check it versus future sessions.
3. Learn a change up. 9/10 years of age is a good year to start learning this. Try different grips until he find one that works. My son went through 3-4 grips before finding the one that works well for him.
4. Learning pitching strategy. The most under-coached part of pitching. Too many youth coaches call every pitch of every game. Most youth pitchers wouldn't second guess a catcher who calls for a change-up after a batter fouls two straight balls down the opposite field because he's swinging late. Create scenarios when your son is pitching to you and ask him what pitch should be next and where.
5. Improve pick-offs. IMO, another under coached part of the game. Teach him to control runners, and not let them control him.