How to fill in the spaces on the floor of wooden deck.

Decks have spaces. They are flat and the spaces let the water drain. If you are using it as a roof, you will have to build it as a roof with an underlayment and roofing material. Roofs have a slant to let the rain water flow off, so you will have to build that into your former deck. BTW: yes, you can still use it as a deck with the changes. Just make sure the material you use is good to walk on and put chairs on.

put a roof over the deck

Depending on what you use the deck for, and how often it is used, a quick fix would be a waterproof tarp covered by an outdoor-type rug. That way whatever drips through will be around the perimeter of the patio below.

Tuftex pollycarb corrugated plastic underdeck roof panel system available at most home improvement stores.

The spaces are for expansion of the decking. If you fill them in the decking might buckle. If your can't put a roof over the deck then try to figure out a way to put one under it.

It is more likely you have a leak somewhere in the roofing over where the rain is coming down. We have a double porch. The upper porch is constructed so that the rain runs off the decking (ours looks just like yours) we knew we were in trouble when it started leaking in the middle of the downstairs porch. Have someone check it for you.

I agree with the corrugated plastic. We had our hot tub under our deck, and in the winter, the snow that was on the deck would melt from the steam coming from the hot tub, dripping COLD drops on us. The corrugated plastic was our answer.