Your request has been heard and we are joining you in prayer for your loved one and for your whole family.
We are praying that the cough and mucus you described will not become anything serious, and that if it is the start of a cold or other illness, the Lord will bring a full and quick recovery. We believe that healing is still the work of God today and that He is able. Sometimes He heals in a moment, sometimes He heals gradually through the natural processes He Himself placed in our bodies, and both are divine because He is the one who made those processes. We can trust Him with her health and ask Him to watch over her with care, to give wisdom for any care she needs, and to bring comfort and strength while she recovers.
We are also praying for the health and goodness of your entire family as you asked. The apostle John once wrote a personal wish for a friend, that he would prosper and be in good health even as his soul prospers. That was not a blanket promise that believers will never be sick, but it does show the heart of God who cares for both body and soul, and it is right for us to bring our physical needs to Him and ask for His mercy and protection over your household.
Above all we are praying for the salvation of your family. The message of salvation is wonderfully simple, and that simplicity is where many stumble because we want to do something great to earn it. Scripture says to believe on the Lord Jesus Christ and you shall be saved, and that Abraham was counted righteous by faith. There is no difference, all can come the same way through Jesus, and He is able to save to the uttermost all who come to God by Him. God has already done the glorious work through the cross and resurrection, but unbelief keeps a person from partaking in it. It is not enough to see or know about what God has done, each person must receive it by faith, confessing with the mouth that Jesus is Lord and believing in the heart that God raised Him from the dead.
There is only one provision God has made for sin, and that is Jesus Christ. Salvation is not by our works, it is by His grace through faith which He gives, and we are His workmanship. When we belong to Him we are brought into one body, the family of God, and that spiritual family is an eternal bond. Our prayer for you and your loved ones is the prayer written for believers long ago, that you would be strengthened with might by His Spirit in the inner man, that you would know His love, and that each member of your family would come to trust Jesus fully as Savior and Lord.
So we pray now, Father, in the name of Jesus, lay Your healing hand upon this dear one who has been coughing, calm any irritation, clear any infection, and restore health. Guard the whole family in body and soul, and draw each heart to Yourself. Open eyes to see the simplicity of Your grace, remove unbelief, and bring each one into saving faith in Jesus Christ alone. Give peace to this family as they trust You, and let Your goodness be evident in their home. Amen.