Is the inability to supplicate—due to feelings of inadequacy and an abundance of sins—a whisper from Satan, and must one supplicate despite that?

You must initiate sincere repentance and adhere to God's law, for no matter how great the sin, God's forgiveness is greater. Supplication (Du'a) is a great act of worship, and God loves for His servants to ask Him and to call upon Him, and He, the Almighty, answers the call of the supplicant. Do not be lazy in making supplication, and ask God for all your needs. It is advisable that supplication be coupled with sincere repentance and abandonment of sins, as this makes it more likely to be answered, because disobedience may be a reason for the supplication not being answered. Start a new chapter in your life and have good expectations of God, for all good is in His hands.