Why does Godfrey take a secret interest in the upbringing of Eppie, and how does he go about doing this in Silas Marner?

Godfrey takes a clandestine interest in the upbringing of Eppie because she is his real daughter, born of Molly, a woman whom he married secretly. Godfrey provides for Eppie by sending things and having things done in and around the cottage.
When Silas Marner appears at the Squire's in Chapter XIII, he has Godfrey's child in his arms. Marner seeks Dr. Kimble, who parts the guests and leaves to examine the woman who has collapsed in the snow. Before he departs, the doctor tells someone to find Dolly Winthrop to help with the child. Godfrey rides out to find Dolly, and to go to Marner's cottage to see for himself if Molly is alive or dead.
When he returns, having learned of Molly's death, Godfrey considers that now is his opportunity to woo Nancy Lammeter. But, he will not forget the child: 

As for the child, he would see that it was cared for — he would never forsake it; he would do everything but own it.

In Chapter XVI Godfrey "did very kindly by the weaver." Among other deeds, he sends the weaver an oaken table and nice beds, and he offers Silas and Eppie some good soil for a garden.One day, after he has had the area near Silas drained, his brother Dunstan's skeleton is found at the bottom of the well that went dry from the draining. He had fallen in after robbing Marner of his gold. After returning this gold, Godfrey goes to Silas Marner's cottage in Chapter XIX and tells Silas Marner how sorry he is that his gold was stolen. Years later, he asks if he may provide for Eppie:

"You'd like to see her taken care of by those who can leave her well off, and make a lady of her; she's more fit for it than for a rough life...." 

When Marner says that he does not understand, Godfrey asks to adopt Eppie. Silas tells Eppie to speak, saying "I won't stand in your way." But, Eppie thanks Mr. and Mrs. Cass and tells them she cannot leave her father: "I couldn't give up the folks I've been used to."
When he realizes that he no longer can take a secret interest, Godfrey tells Silas and Eppie the truth. But, this has no positive effect for him. Eppie insists upon living with Silas, who has been a real father to her.