* U.S. soybean crop rating improves less than expected after rain

* Traders see USDA report cutting soybean area, yield forecasts

* Wheat rises as Odesa drone strike revives Black Sea war worries

PARIS/SINGAPORE, July 11 (Reuters) - Chicago soybean futures rose for a second session on Tuesday as rain gave only a modest boost to Midwest crops, adding to expectations that the U.S. government will reduce its harvest forecast.

Wheat climbed as a drone strike on Ukraine's Odesa port put attention back on war risks to Black Sea supply, while a surprise drop in a U.S. spring wheat crop rating underscored mixed harvest prospects in the northern hemisphere.

Corn tracked strength in wheat and soybeans, but was capped by a sharp improvement in U.S. crop conditions and harvest progress in Brazil.

Just 51% of U.S. soybeans were in good-to-excellent condition as of Sunday, U.S. Department of Agriculture data showed on Monday, up from 50% last week but below an average analyst estimate of 52%.

The limited impact so far of showers in drought-affected Midwest fields supported expectations that the USDA will lower its forecast for U.S. 2023 soybean production on a monthly supply and demand report on Wednesday.

The USDA's weekly good/excellent score for corn rose to 55% from 51% last week, beating an average expectation in a Reuters survey at 53%. However, the rating is the worst for this time of year since the drought year of 2012.

"U.S. weather over the next few weeks is key to determining soybean and corn yields and can improve production prospects," said Pranav Bajoria, a director at Singapore-based brokerage Comglobal.

The most-active soybean contract on the Chicago Board of Trade (CBOT) was up 1.3% at $13.62-1/2 a bushel by 1128 GMT while corn rose 1.2% to $5.05-1/4 a bushel.

CBOT wheat added 1.8% to $6.58 a bushel.

A Russian drone attack on Odesa on Tuesday caused a brief fire at a grain terminal near the city's Black Sea port, Ukrainian officials said.

While the damage was marginal, the news highlighted uncertainty over the future of a Black Sea grain deal, which Moscow is threatening to quit next week, traders said.

The decline in the USDA's spring wheat crop rating added caution about global milling wheat supplies despite ample stocks in top exporter Russia.

(Reporting by Gus Trompiz in Paris and Naveen Thukral in Singapore; editing by Sonia Cheema and Jason Neely)