1. Thoroughly wash the blackberries under running water to ensure they are clean.
2. Place the blackberries in a small saucepan.
3. Add 50 milliliters of water to the saucepan with the blackberries.
4. Bring the mixture to a gentle simmer over low heat and cook for about 5 minutes until the blackberries are soft.
5. Remove the saucepan from heat and let it cool slightly.
6. Transfer the softened blackberries and water to a blender or food processor.
7. Blend until you achieve a smooth puree. You may add a little more water if the puree is too thick.
8. Pass the puree through a fine mesh sieve or strainer to remove any seeds to obtain a silky texture.
9. Allow the puree to cool completely before serving it to your baby.

Summary:
This Blackberry Baby Puree is an easy and nutritious snack suitable for babies aged 4-10 months. It is rich in vitamins and antioxidants, which are beneficial for your baby's growing body.

Storing Options:
You can store any leftover puree in an airtight container in the refrigerator for up to 48 hours. For longer storage, the puree can be frozen in ice cube trays and then transferred to a zip-lock bag for up to 2 months.

Possible Side Effects/Allergies:
While blackberries are generally safe for babies, they can occasionally cause allergic reactions. It's always best to introduce new foods one at a time and watch for any reactions, such as rash, hives, or digestive upset. Additionally, the small seeds in blackberries might be difficult for some babies to digest, which is why passing the puree through a sieve is recommended.