Sparse regression such as the Lasso has achieved great success in handling high-dimensional data. However, one of the biggest practical problems is that high-dimensional data often contain large amounts of missing values. Convex Conditioned Lasso (CoCoLasso) has been proposed for dealing with high-dimensional data with missing values, but it performs poorly when there are many missing values, so that the high missing rate problem has not been resolved. In this paper, we propose a novel Lasso-type regression method for high-dimensional data with high missing rates. We effectively incorporate mean imputed covariance, overcoming its inherent estimation bias. The result is an optimally weighted modification of CoCoLasso according to missing ratios. We theoretically and experimentally show that our proposed method is highly effective even when there are many missing values.